You've lost me on question #1. Independent contractors aren't eligible for UI. You were recharacturized as an employee because they felt you WERE an employee (for this you should be lucky. Note you should also go back and amend your tax returns for prior years and file to recover the share of the SE tax you paid that should have been covered by the employer). Once you're been determined to be an employee, then they need to see if your termination was due to your own misconduct. I'm not having a lot of luck deciphering the sequence here (especially with your failure to submit timely appeals, hopefully you got a ruling that the delay was for due cause).
You can't just take things to the Supreme Court after some administrative board rules against you. You need to pursue whatever administrative appeal process there is and THEN you might be able to bring it to Superior Court for action.
The Connecticut DOL web site seems to have plenty of information on the process. A lawyer may be able to keep you from screwing up though.

