Good morning Yosef,
Here is a solution:
1. Take your inspection checklist and have the contractor complete the work he's already been paid for, make that part of the mediation.
2. I wouldn't use them to complete the job, they've already shown they aren't doing quality work. I would also look at suing them for the cost / expense to finish the job.
3. I wouldn't pay them anything upfront...I recommend you always pay for labor AFTER the work is done. Pay for supplies upfront.
4. Hire an inspector /independent person to manage the job site for you, someone with your best interest at heart. Make it clear that you'll only pay when that person signs off on work done. Ensure your agreement with that person has teeth so if they sign off on something that isn't right they can be held liable for it.
Answers to your questions:
1. Try mediation first, lawsuits can get expensive and only the lawyers win.
2. Removing a lien is challenging because you need the person who placed the lien to sign off to remove it. I'd make this part of the mediation process. Part of whatever agreement is they can't place a lien. I'm willing to bet you may have to pay some of their demand to ensure this doesn't happen.
3. Have 2-3 contractors look at the job and get written estimates on what it's going to take to finish the job, correct the mistakes, pass occupancy certification etc. use those estimates in your countersuit / mediation process.
4. This sounds like it's going to be an expensive lesson.
a. You started off good with cost being paid in stages, but it sounds like you paid before you verified they actually hit their benchmarks.
b. Hopefully you can show where he didn't honor the agreement that cost changes had to be approved...hopefully that approval was supposed to be in writing. He may have to eat those additional expenses if he can't prove he had approval prior to doing it.
c. Consult a good real estate attorney.