How can God forget our sins if we can remember them? Doesn't he know what we are thinking? How can he heal us from things if he doesn't remember them?
How can He  forgive us if we remind him by thinking about them? How can we be honest and transparent if we shouldn't remember stuff in order to not remind God of the stuff he has forgotten?
Wow, thanks for trusting  me with this wonderful and wrestling series of questions. Yes, "God knows all things" (this is simply quoting 1 John 3:20); so He knows what we are thinking at all times; and He knows what went on in the past, including our forgiven past.  Let me start by offering short concise answers (amazing for a preacher, huh?) on the first and last questions, and then spending some time and type in explanation, which will hopefully address your whole package.
How can God forget our sins if we can remember them?
Contrary to a popular interpretation of Scripture, the Word does NOT anywhere say  that God forgets our sins.  (However, if He did, it shouldn't be a problem for Him to "forget" them, even if He allowed us to remember them.  How could He pull that off? I don't know, but He's God, and if His ways required Him to do that, He could do it with integrity..  But again, I do not believe this is the scenario Scripture paints) . He does not at all "forget" our sins, or perhaps I should say at least in the way we currently understand that word.
How can we be honest and transparent if we shouldn't remember stuff in order to not remind God of the stuff he has forgotten?
Clearly, the Spirit calls us to be (though we don't always "go there"), as you say, honest and transparent.  I believe so passionately and intensely in honesty and transparency that our congregation has from day one, cherished these  as core and uncompromisable values.  I DON'T believe that we should  (again, this is contrary to so much teaching today, but I believe it is truth) put pressure on ourselves to "completely forget" our past and forgiven sins.  So the answer to the question, the way you have poised it is: We can't.  But the good news is: (to quote a great book title from Larry Crabb...
The pressure's off! We are not asked to do so.  Sure, we need not dwell or obsess on past sins; and we probably never need to intentionally call them into memory or mention them.  But
God does not get ticked off at us, if we occasionally "dig up" (in conversation or prayer) old and forgiven sins of ours; only lovingly sad that we haven't grasped how complete forgiveness is.
 I confess that, in my preaching and teaching, I have surely perpetuated the myth that "God forgets our sins."  It has just been too "preach-able."  But technically, and very significantly, God does not ever claim that this is how He handles forgiven sins.  He suggests not that He has forgotten our sins, but that He "remembers them no more."  Yes, these are two different things; and yes, this is  the phrase in every Scripture which we "thought", or wanted to believe, used the word "forget".
Examine these Scriptures:
  •         "I am the Lord, who blots out your transgressions....and remembers your sins no   more." (Isaiah 43:25)
  •         "I will forgive their wickedness..and remember their sins no more." (Jeremiah 31:34 and Hebrews 8:12)
  •         "Their sins and lawless acts I will remember no more."  (Hebrews 10:17)
    Alright, having established that distinction in vocabulary, the obvious question is: How are "forgetting" and "remembering no more" to be distinguished?  Aren't they simply different angles on the same reality?  No.  Let me cite a great and helpful article from the Christian Apologetics and Research Ministry:
God is omniscient (1 John 3:20). That means that He knows all things. If this is so, how could he not know about our sins if the Bible says He doesn't remember them? Actually, the question isn't entirely correct. You see, not remembering and forgetting are different. You cannot forget something on purpose, but you can purpose not to remember something. You don't make yourself forget where your car keys are. You don't say, "Okay, I'm going to forget where my keys are…uuhmph!.. THERE!. Now. Where are my keys?" Forgetting is something that happens to you. Remembering, on the other hand, is something you choose to do. You can remember where your home is, what your dad's name is, where you work, etc. You actively bring to your mind what you need to know. That is called remembering.
     God chooses not to remember our sins. He chooses not to ever bring them up again. He doesn't forget because He cannot. But, He doesn't remember them either; that is, He will never bring them up again. Why? Because they have been forgiven in His Son Jesus.
     This is why, when you read Hebrews 11, you will see a listing of great people of faith. Abraham, Moses, David, etc. Yet, none of their sins are listed even though they were sinners. Why? Because God remembers their sins no more.
     One other point: Jesus told us to forgive as the Father has forgiven us (Eph. 4:32). So, if someone does something wrong against you, then later asks for forgiveness, and you forgive him, you are never supposed to bring up that offense again… ever! Maybe you won't ever forget the offense, but then again, you can choose not to remember it. This is what forgiveness means for those of us who have been forgiven by Jesus.
Helpful insights.  But crucial, and deeper to the heart of the argument, is what exactly does the biblical Hebrew word "remember" actually mean?  This is the arena where we often trip up in grasping and applying Scripture; not taking time to consult a highly accurate translation or paraphrase, and finding (quite often to our surprise or shock) that in the language which the Lord chose to have His Book written in, terms do mean different things than they denote or connote in English.  A classic example is the term used in the verses catalogued above; a term we usually translate "remember," but fundamentally means "to allow something in the past to affect us today.".
This is why God throws down the challenge so many times in His Word (Exodus 13:3, for example) to "remember" what he has done, and done for you.  This is not a lecture, as in "Don't let this fact slip your mind, dummy;  Write it down!"  It's far more of an "In light of what I have done in the past, be energized by the exciting  potential of what I am doing, and can do,  in you right now!  In fact, what I have done in the past for you is still impacting  you this very second, and will impact you even more if you 'remember' it."  To "remember" also implies  to, in a  sense, actually participate in or draw from the past, and to "let it affect you profoundly now."  Clearly the most central example of that for Christ-followers is the Cross. It is  a historical; event that happened 2,000 years ago.  But by faith,  "we were there when they crucified the Lord", even though none of us reading had even been  born then (If you were, e- mail me!  I want to interview you!).  Does Calvary affect me today?  To the core!  That's the whole point of Christianity!
Read carefully, and soak in prayerfully,  the words of Dr. Victor Shepherd of Tyndale Seminary below:
Think of the word "remember." It doesn't mean in English what it means in Hebrew. What "remember" means in Hebrew we shall grasp more readily if we think first of the difference between Hebrew and English meanings of the word "forget." To forget, in English, is simply to have an idea or notion slip out of the mind. To forget a person is simply no longer to have the idea of that person in one's consciousness. But in the Hebrew bible to forget someone is to annihilate that person, obliterate him, destroy him. When the Israelites cried to God not to forget them they didn't mean, "Be sure to think of us once in a while." They meant, "Don't destroy us, don't annihilate us, don't blot us out!" It's obvious that to forget, in Hebrew, has to do not with ideas but with living realities.

In the same manner to remember has to do not with recollecting notions but with living realities. In a word, to remember, Heroically, is to bring a past event up into the present so that what happened in the past is the operative reality of the present. To say the same thing differently: to remember is to bring a past event up into the present so that what happened then continues to happen now. What unfolded back then, altering forever those whom it touched, is to be operative now, altering forever those who "remember" it now. When the Israelites are urged to remember the deliverance from slavery of their foreparents centuries earlier they aren't being urged chiefly to recollect a historical fact; they are being urged to live the same reality themselves hundreds of years later. Just as their foreparents knew most intimately a great deliverance at God's hand, together with the gratitude and the obedience which that deliverance quickened, so they are now to know most intimately a similar deliverance at God's hand, together with a similar gratitude and a similar obedience.  (source:.p: //

 We might summarize and finalize our working definition of what the Hebrew language, and thus the Lord, had in mind by the astounding claim that He "remembers our sins no more" by stating that  "to remember" It means we to "allow something in the past to shape who we are and what we do in the present reality."

The qualifier! Despite what I said earlier, it is in a way true that God forgets our sins.  But again, we don't mean that in the sense of our English language.  To "forget" in the Bible language, culture and mindset, UNLIKE OURS,  does not entail losing or dropping something from our memory bank, or the condition we all know of something  "slipping our mind" (forgetting an appointment, or not remembering to bring keys or wallet).  In fact, in a certain sense, an opposite meaning is implied.  Paradoxically, to "forget" in the Bible is to REMEMBER, but not allowing the memories of the past to impact or influence (especially in a negative way) our present or our future.  For all practical purposes ,in the sovereignty of God, our past--even though He and we can remember much of it-- does not exist; or is not allowed to affect, detour or deter our current walk! Our past sins are  "annihilated, obliterated, destroyed"..ultimately this happened on the cross when Jesus "became sin" on our behalf through the "annihilation"  He surrendered to then and there.  So in this focused  sense, God does "forget" our sins..but this doesn't mean we have to imagine or theologize about an Omniscient Being who has somehow developed selective amnesia; it means He has voluntary chosen not to hold the knowledge of your forgiven sins against you.  He won't even bring them up anymore.  It's almost as if they have been erased from His memory bank, because they have  been, in essence  "annihilated and destroyed"  Or as the Bible pictures in Psalm 103:12, "thrown into the far as east is from west...where they will be remembered no more"  And Corrie Ten Boom has suggested that  He has  posted a prominent "No Fishing" sign on that sea.  Does  that necessitate that we, or God, is completely amnesiac  about actual events in the past?  No.  (Psalm 139 reminds us that God is not only omniscient. that is, knowing everything...but also omnipresent. present everywhere at once. so He "knows" where the east and west seas ..and thus your sins... are). But He does not hold them against us,  He does not "go there", even though He must know where the "there" is. They are buried and therefore "stricken off the  "judgment  day" record against us, and if we do "remember" them (in the English sense of the word,) it should not lead us into heart-piercing shame but heart-lighting praise.  "I clearly remember forgetting that", is what the Lord says in one popular (non-biblical) story.

   Another area in which we Bible-believing people have put far too much pressure  on ourselves  in  is the thinking  that  we must "forgive and forget."  Even though many would claim "forgive and forget" is commanded in the Bible, it is not.  An even better deal is to remember and still forgive; to choose not hold anothers'  sins against us against them.  This of course is the fantastic and flooring  thing that  God has done for you...

Don't torture yourself by giving into the temptation of believing you have utterly failed because you "remember" your own or another's sin.  The torture was already endured by a gracious Christ on a gracious Cross!

Pray that I don't "forget" it!