WHAT TO DO IF I GET SUMMON/LEGAL NOTICE FROM LOWER COURT FOR DIVORCE?

The best thing you can do is consult with an attorney and obtain representation asap. You have much work ahead of you.

Can I file divorce in MA and would my case be strong if I relocated to NJ?

You really need to speak to attorneys licensed in both states and then you can make some sort of determination about which state is more friendly...

DDo I have to pay child support if we're still legally married?

Yes you still will be on the hook until a new order/agreement supercedes this.

Can I argue in the probate court about the prenup agreement ?

its hard to understand what is going on here. Please speak to your attorney.

Can we travel in-state (NY) with noncustodial children without custodial parents permission?

Something does not sound right. You might want to seek a consult with an attorney and discuss what your options are.

Is a notarized handwritten document still valid if circumstances have changed?

Its possible or not depending on the validity of the document and what is in it.

How to take my ex wife of 17 years off my Coop Apartment Deed?

You need someone to review your divorce agreement and then investigate what needs to be done further.

After being placed in domestic violence shelter do I receive custody rights to child?

No. You need to petition for custody in Family Court.

You really need to speak to your attorney and find out why its not going well, and what you can do now or in the future to make it work.

What will happen if I can prove that my son's father lied about the reason he wants full custody?

Without any more information, its impossible to even guess an answer. I strongly suggest you to please hire counsel to assist you in your action.