Hi Ariel,
You have a couple of great questions, which should make us think and ponder what scripture says on this. I can only give you what I believe scripture says - we have to bare in mind that all major branches of Christianity believe in 'original sin', I have not written out scripture but have given book/chapter for those who wish to have a study themselves.....it will pay of if you do. thats my prayer anyhow.
Q. Was Adam created with a "sin nature" or was sin placed upon him when he disobeyed God?
A. God created all things and said all that he made was good (Gen 1:31). Our first parents (Adam & Eve) were created with original righteousness and communion with God. But through their sin of eating from the tree of Knowledge of good and evil sinned against God. And therefore spiritually died, they died in their sin (Gen 2:17; Eph 2:1). Sin here is not just an outworking or an action based on ones own heart, it is wholly rebellion against God and affects every part of man -spiritually and physically (Gen 6:5; Titus 1:15; Jer 17:9; Rom 3:10 -19).
Adam and Eve were created good and righteous, but because of their sin, we have the Fall (Gen 3). This fall corrupted the 'image of God' in human nature - it has been blurred!). Adam and Eve from this point on had the knowledge of evil, their desires had changed from being righteous and in communion with God, to rebelling against Him in their hearts and minds.
That's the nuts and bolts of your first Question which leads nicely to the next one.
Q. Are we born with sin or does our hearts become corrupt when we fall to temptation and disobey God?
A. Adam and Eve being the 'root' of all humans, through them the guilt of this sin (rebellion/heart) was imputed to all humans (Gen 1:27-28; Gen 2:17-18. cf, Acts 17:26; Rom 5:12-19 and 1 Cor 15:21,22,49. All humans beings descend from Adam and Eve so share in the same death in sin (Eph 2:1) and corrupted nature (Ps 51:5; Gen 5:3; Job 14:4 and 15:14).
From this original corruption (original sin) our hearts are always inclined to evil and opposite to that which is good (Rom 3:10,11,12). See also Gen 6:5, many will try and tell you that this is only relevant from before the flood but carefully note Gen 8:21 after the flood. It is from this original sin (corruption) that proceeds all our actual sins (James 1:14-15; Matt 15:19).
So every sin both original and actual is a transgression against the law of God (1 John 3:4) and this very nature brings guilt on the sinner (Rom 2:15). Which subjects the sinner to the wrath of God (Eph 2:9), the curse of the law (Gal 3:10) and death (Rom 6:23).
"The doctrine of original sin teaches that people sin because we are sinners. It’s not that we are sinners because we sin, but rather, we sin because we are sinners; that is, since the fall of man, we have inherited a corrupted condition of sinfulness. We now have a sin nature. The New Testament says we are under sin; we have a disposition toward wickedness, so that we all do, in fact, commit sins because it is our nature to commit sins. But that’s not the nature that was originally given to us by God. We were originally innocent, but now the race has been plummeted into a state of corruption. (R C Sproul)"
"[so] When we speak of the fall and of original sin, we are not speaking of the first sin committed by Adam and Eve, we are speaking of the radical consequences of that sin, which followed to all future generations of mankind (R C Sproul)."
I have not discussed the good news of salvation in our Lord Jesus Christ as the questions were specific, and also the title of the OP was specific also. If we get this wrong we get salvation wrong. The thought that people are born neither good or evil, or born innocent cannot be found anywhere in scripture this is the view of pelagious and our current culture. As Jonathan Edwards notes if all people were born innocent you would expect atleast or up to 50% to to grow and never sin (RC Sproul quoting Edwards). We all know just by our own experience and looking at the world this is not true.