Yes to the first part (Ps. 51:5; Rom. 5), but to the second part...not exactly, as it's not really relevant. The problem is our very nature has been "made opposite unto all that is spiritually good," such that "actual sins" of individuals will inevitably be the fruit of that corrupted root. Here's how we put it:
```
WLC
Q. 22. Did all mankind fall in that first transgression?
A. The covenant being made with Adam as a public person, not for himself only, but for his posterity, all mankind descending from him by ordinary generation, sinned in him, and fell with him in that first transgression.
...
Q. 25. Wherein consisteth the sinfulness of that estate whereinto man fell?
A. The sinfulness of that estate whereinto man fell, consisteth in the guilt of Adam’s first sin, the want of that righteousness wherein he was created, and the corruption of his nature, whereby he is utterly indisposed, disabled, and made opposite unto all that is spiritually good, and wholly inclined to all evil, and that continually; which is commonly called original sin, and from which do proceed all actual transgressions.
Q. 26. How is original sin conveyed from our first parents unto their posterity?
A. Original sin is conveyed from our first parents unto their posterity by natural generation, so as all that proceed from them in that way are conceived and born in sin.
```
Regarding the question of free will and predestination, I wonder if this might help. My favorite analogy to explain this is that of a magnet compass. A magnetic compass is attracted in a certain direction, by its very nature, which nature it is powerless to change of itself (nor does it want to). The curse, in effect, 'reversed the polarity' of all humans 'descening from Adam by ordinary generation,' such that rather than being attracted to the glory of God, it becomes repellent to us. Our 'needle' points South because of that switch in polarity. We hate the idea of a supreme judge who will hold us accountable ("for our deeds are evil"), we repress this knowledge in unrighteousness (Rom. 1), etc. The needle spins _freely_ -- as in free from some outside force -- but that needle is not free from its natural magnetic properties -- it turns where it wants to turn, by nature. In regeneration, the grace of God soveriegnly and supernaturally _reverses the polarity back_ such that we freely spin our 'needle' back to true North because we now want to. The teaching of predesination is not that God reaches down and grabs hold of that needle and forces it back toward himself--it's that he supernaturally changes our very nature, and we--now truly free--'spin our needle' back to him. Freedom of the will does mean freedom from outside force, but it does not mean freedom from our own nature.
A man is completely free to choose whatever his nature desires; and we always choose what we want most. What, then, will a "dead," "blind," "enemy" choose freely according to his nature? All that is not God. The problem is not our freedom, it's our nature. "We must be born again from above."