You may want to consider filing a complaint with the state (https://www.myfloridalicense.com/entercomplaint.asp?SID=)
Your contractor came in on an emergency job, bid $4,000.00 and now is threatening legal action to charge you 3 times that much.
They refused to answer your repeated calls regarding how much was billed, and now they want to file a lien or take yout o court.
I would go to the contractor's board. (The link above tells you how the process works, and what the state's role is).
As far as the lien against your home, etc. - if you aren't worried about the equity in your home, I wouldn't worry about a $12,000.00 lien (they aren't going to try to foreclose as a junior lien holder on an underwater home, the lien will just sit there until the home is sold or refinanced).
But if they sue you, make sure that you answer the complaint and go all out on a defense - these guys are taking advantage of you.
- but again, start with the FL State Complaint (I have a strong suspicion these guys are taking advantage of you, and probably violating a lot of state contracting laws).