Can a general contractor file a mechanics lien if they use unlicensed subs and did not complete written contract?

They can file the lien, but in this case, I think they can be persuaded to remove it, with a good stiff attorney's letter.  They didn't comply with the contract, so they can't enforce the full price.  In fact, it seems to me that you'd be justified in having the roof re-done and charging them for the work.  Their opinion on whether or not the existing fan was good enough is irrelevant;  if you were being charged for something that wasn't installed, they can't collect for that.

And, as the general contractor, it was their responsibility to ensure that the subcontractors were licensed.  If you can prove otherwise, it's very much in the general contractor's interest not to have the unlicensed work brought to the attention of the building inspector.  You have quite a bit of leverage here, and I'd recommend talking to a lawyer, to find out what all of your options look like.