Will they take arrears at next court date?

Verbal agreements do not lead to arrears. The judge will decide appropriate support levels dating back to the filing of the petition seeking support.

How do I get a religious annulment?

From whatever church is involved. Each has its own rules.

Logjam... and a very big one. It takes a very long time for things to get posted.

Once judge signs judgement or divorce papers how long will it be processed by county clerk for entry & can it be expedited?

That depends upon the county clerk's office and its procedures. Typically it's under three weeks... except in the Bronx, Queens and Brooklyn, where...

Can I file for a annulment after I sign divorce papers I have been served with?

No. Once you're divorced there is no annulment - at least in the civil sense. Religious annulments are a different discussion.

Child support

It's unlikely that there would be a paternity test under those circumstances.

How can I receive a complete divorce record?

Divorce records are maintained in your county's courthouse.

If you get an order that says you have exclusive use and occupancy you can change the locks.

Do I need a attorney

Yes, you'd do well to have an attorney...and whether she gets half of anything depends upon factors not included in your post.

Can a lawyer hold on finalizing a divorce because the client he is representing is not paying him?

If it's uncontested there aren't "ends" - it's filed or it isn't. If it isn't, it's contested and your lawyer should proceed accordingly.