Yes, they can require this of you or terminate you if you do not want to take on the different position/role.

There does not appear to be a claim here that you would be able to pursue, unfortunately.

What rights do I have? NYSDOL appeal? Discriminated for being a working mother?

You should file an appeal of the DOL determination as it seems pretty clear that you will be ultimately deemed eligible for the benefits. As for...

In short, you can't be forced to take a position, but they can fire you if you decline to do so, as NY is an employment at will state.

Possible retaliation and or discrimination action against NYCDCAS

If you believe they violated rules, procedures, or the law, you may seek to file an Article 78 petition challenging their actions.

You certainly will need an attorney to assist you with this and, at least as an initial matter, to review the non-compete and discuss the situation...

Yes. You should not forge a dr. note. Why not just get one from your dr?

What is the deadline for making a claim for non-payment of unemployment benefits due to someone else's technical error?

I'd recommend asking for the supervisor's email and communicating via email so you have a paper trail of your efforts.

How do i get in contact with the judge ?

You can probably just write a letter to the judge, and I would copy the attorney on the letter.

I am sorry you are dealing with what you are. You absolutely should contact an experienced employment attorney to have them help you with this...