Question: "What happens to man’s free will in heaven, since free will was the cause of man sinning?”
Answer: Thank you for your excellent, thought out question! First, it will be necessary to clear up some misconceptions about God, who is of an entirely different order than man. The following words may not seem relevant to your question, but they will get us to where we need to go if you bear with me.
We can understand certain things about God from nature. All that we see was created in time. It is dependent upon something else for its existence. But there cannot be an endless regression of causes for existence, so there must be a self-existing Being upon which all existence depends. This is God, of course.
Every attribute that the created order possesses must first exist in a higher form in the God who created it. (Evil is not an attribute, but a privation of a good attribute that ought to be there.) So, any attribute that a created thing has must be brought into actuality by a prior cause. But there is nothing before God upon which He depends to become what He is now. All that He is, He is from eternity. God cannot change (Ps. 55:19, Num. 23:19). If God is love, He is love eternally. If He is joyful, He is so from eternity. God, who exists eternally in three persons, has need for nothing else. He does not create for His good, but for the good of creation. He does not need humanity at all. We need Him. An infinite being cannot change because nothing can be added or subtracted from an infinite.
Now, since God is fully actualized (He is the same from all eternity), it is impossible that He could violate His own nature (sin). But man is not like God in this sense; man can change. For God, freedom is having the power to do that which pleases Him, and it is impossible for Him to do otherwise because of His nature. Not so with created things. Humans can actually choose evil without it being contrary to reason. So, we can't have free will while being guaranteed not to sin, "modeled after the characteristics of the Trinity," as you say.
What if God put His creation in heaven to begin with? He did, and the angels fell (Rev. 12:9, Luke 10:18). They fell because they became obsessed with their own beauty and power rather than the ultimate power that they were made to know. Likewise, Adam wanted to "be like God" and decide his own fate (Genesis 3:5).
So, why couldn't God make man in a way that it would be certain that he would have freedom but not rebel? Such a state of affairs may indeed have been impossible. Again, it would not be possible for man's freedom to be the same as God's, as I described earlier. Man has the potential for pride, to look to himself rather than God for fulfillment. As long as that is true, he has the possibility of sinning.
What about in heaven? Isn't it true that in heaven, people have freedom but do not sin? Yes, but people in heaven have knowledge of sin that Adam did not have before he fell. In heaven, the redeemed will be able to fully know the contrast between God's goodness and the horrendous effects of evil. With glorified minds and bodies, they cannot possibly choose evil. Could God have given man that knowledge beforehand? God did give Adam the warning (Gen. 2:17), but mental knowledge is different from experiential knowledge. Having experienced the full effects of sin and the presence of God, one will not decide to turn back to sin.
In His wisdom, God sees it best to give man what he wants without fully knowing the consequences beforehand. Perhaps this way honors human dignity and gives Him the opportunity to express His great mercy and love. We can more fully understand God's love and His conquering power in light of a history marred by sin. We do not know fully why God does as He does, but we know He is good (how we know that must be accounted for elsewhere). This is obviously not the best possible world, but, as my mentor Norman Geisler has often said, it may be "the best possible way to the best possible world."
As for your final objection, that a rule exists which God does not decree: the laws governing existence are not from outside of God, nor are they merely decreed arbitrarily as you suggest. They derive from God's very nature. Order and logic are derived from God's Being. A state of affairs in which man has freedom and yet cannot sin is contrary to reason, as I have argued here. And reason is from God, not from outside of Him. So, there is no obligation on God from the outside. He is Sovereign, and in His sovereignty, he has allowed evil, which He will ultimately use for His glory and for our good.
Praise God for His wisdom, power, and love!