Okay, regarding unemployment insurance. File a claim for benefits right away. NOW. Then they will go through this whole business, sift and sort, follow the laws of your state, look at your company policies, your individual situation, and determined whether the company had a valid misconduct reason to terminate.
It sounds as though you are dealing with an overly fire-happy H.R. at your job, because you don't have a clue, and it sounds as though some of the things you have picked up from your HR is not accurate. If they are in the business of trying to cut their higher paid staff, they are lying in wait for reasons they can fire you. And yes, they have the right to do this. It's legal to fire employees, with or without a good reason.
But they certainly don't get to say whether or not you get unemployment benefits. What you do is file the claim, telling the exact verbage you were given and what happened to you the last day you worked, or didn't work as this case may be. The unemployment system will then look at the employer's policies, what they received from the civil authorities, your past attendance, all these factors and determine if the company had a valid misconduct reason to terminate you.
You do not take this company's word for anything at this point. They are not your friends, and will not tell you anything straight concerning unemployment or your rights. This general release they are asking you to sign is another thing that sounds like an over zealous HR folks who think they're smart. You can not sign away your rights. (COBRA coverage, minimum wage, unemployment benefits, right to sue if EEOC rights are violated)
Those types of agreements may make you think you can't get this, do this, or file this, but actually, you still can, no matter what you've signed. And though they threaten, I don't think they can take COBRA away from you based on this termination, whether you sign the release or not. But I'm not seeing any downside to signing any release. I don't think you would have any grounds to sue the company anyway that I'm hearing. You'll go on and file for your unemployment benefits at once, and the wage and hour division of your state will probably be the ones to talk to about COBRA coverage. You other folks correct me if I'm wrong about that, I'm not really a COBRA expert.