Get yourself a good lawyer now. Maybe Liz can point you in the right direction. I'm not a lawyer, so this is not legal advice. However: If the lien has been satisfied, getting it removed sometimes can be a bit of a hassle, but ultimately not a problem. You have to get the previous lien holder to provide a certificate of satisfaction (or similar terminology) that is recorded to get the lien removed. You or your lawyer should be able to handle that without much problem.
A quit claim certainly is valid if the person giving it has anything to give. I could give you a quit claim deed to the Statue of Liberty. It would be entirely valid, legal, and enforceable. Problem is, I don't own the Statue of Liberty. So that quit claim deed would be worthless. On the other hand, if I gave you a quit claim deed to my personal house, you'd now be the owner of my house.
And from your other questions, you're way in over your head. Again, I'm not a lawyer, but you record the deed, just as you would a general warranty deed. From what you're describing, the mortgage company is totally out of the picture. Ownership went from previous owner to mortgage company to second company to you. You get title in your name by making sure you've got the correct documents and filing them.
You say the only one you talked to was the lawyer representing the second company, from whom you bought the property. Hate to break this to you, but that lawyer was representing the other side. Not you. You didn't have any representation? In any case, that lawyer's correct. He was representing the second company. The transaction was completed. He has no further role and certainly no relationship with you.
Right now, you're just twisting slowly in the wind. Possibly, everything was above board. Like I said, removing that lien, if it was really paid, shouldn't be too much of a problem. If the second company owned it, the quit claim deed should pass ownership to you. But you desperately need an expert to sort through all of that and to protect your interests.