I learned from the Quran that our first parents, Adam and Eve, were living in Paradise until they committed sin/wrong. Quran says that they were forgiven of their sins/wrongdoing. If they were forgiven of their sin/wrong, then why did God being so merciful and gracious make them leave the garden? Does it mean He did not forgive them?"
There were four sequential steps in Adam's story:
[ 2:35 ] And We said, "O Adam, dwell, you and your wife, in Paradise and eat therefrom in [ease and] abundance from wherever you will. But do not approach this tree, lest you be among the wrongdoers."
[ 2:36 ] But Satan caused them to slip out of it ...
Qortoby Tafseer states that this slip is considered a sin (Arabic reference: here). That sin caused them to be punished.
[2:36] and removed them from that [condition] in which they had been. And We said, "Go down, [all of you], as enemies to one another, and you will have upon the earth a place of settlement and provision for a time."
- Repentance, Acceptance and Mercy
[ 2:37 ] Then Adam received from his Lord [some] words, and He accepted his repentance. Indeed, it is He who is the Accepting of repentance, the Merciful.
Before God blew his breath into Adam (a.s) he told the angels 'this creation of mine shall be my emissary khalifa on earth and I shall place generation after generation of them on earth'. So no, Adam (a.s) was not sent to earth as a punishment, rather it was meant for him to be there before he was even given life.
If you have any other queries just post them and I will try to give you an answer to the best of my knowledge. The answer I have given you is from Quran.
Since they come to earth, we have chance to show our good deeds. Since heaven is not the place of examination whether a person is good or bad, we, human beings, didn't have chance to show our work. Therefore, by making
Adam and Eve(radiallahu anhum) leave heaven, Allah didn't become unmerciful, in a way, he become merciful to his believer creatures. Even we can say Allah was merciful to
Adam and Eve in that event.
Prophet Adam and Eve did their best in the life and he is named as Safiyyullah(one of the cleanest creature of Allah)
My explanation is that It was God's master plan To send both Adam and Eve to earth. God Put them in a test by telling them not to eat a certain fruit and God knew that Adam and Eve will not follow his order. As he was merciful He forgave them when they repented but ordered them to leave paradise as by eating the forbidden fruit they had developed sexual Feeling and were deemed unsuitable for dwelling in paradise. So He ordered them to go to earth so that they can reproduce and perform more good deeds. Offcourse satan would stand in Their ways to do that but it was a test which God Believed out of His love for human beings that They must eventually overcome. That is why God said when Adam and Eve were sent to Earth that His mercy shall prevail over all wrath. Dr Haider Mannan
Allah's plan was to let Adam and eve live in earth in the first place, the sin was just to show that the satan is our enemy:
And [mention, O Muhammad], when your Lord said to the angels, "Indeed, I will make upon the earth a successive authority." They said, "Will You place upon it one who causes corruption therein and sheds blood, while we declare Your praise and sanctify You?" Allah said, "Indeed, I know that which you do not know." [2:30]