My mom, 86, is being sued for a credit card. Is she judgement proof?

Age is not a relevant factor. You should definitely speak with a lawyer to make certain that she is "judgment proof."

If a judge put a letter into evidence -do I have a right to get it back the Nassau County Supreme court is claiming they do not

Evidence may be discarded after 30 days if left unclaimed.

In the State of New York, can a defendant give his written Verified answer to a served and written Verified Complaint

What you are requesting is acceptable.

Can my neighbor cut part of my tree so he can install a fence?

Yes he may.

If you are at the stage where a Note of Issue is required and you don't know what to do, then perhaps it is time to consider retaining a lawyer.

I was forced to open up a bank account and it still looks bad on me because i was not threatned by gun to open it .

Opening up a bank account does not look bad. Did you authorize the phone company to bill you for the phone you don't have? If so, you should pay it.

If my landlord did not keep my security in a separate acct what action can i take against him

Is the money still there? If so, instead of being litigious, why don't you remind him to do so. If the money is gone, then sue him!

I was mailed a income execution from the city marshall stating the creditor (a bank) has a judgement against me?

The income execution should have the name of the law firm and you should also have received a notice from the Sheriff or the Marshal. Contact both...

Does the condo board have the right to charge me an administrative fee of $1000 yearly because I rent my unit ?

Look at the by-laws and if you don't like them, run for a position on the board, or make a motion to the membership to amend.

Can a motion for a Court Order be included in a Answer/Counterclaim? If so what is the proper form?

Answering the the first question only, the answer is "No."