In New Jersey, Legally can a parent, spouse or friend (moral support) be present during a deposition?

Not unless there's an agreement for him or her to be in the room.

You can not avoid sitting for the deposition yet the location needs to be local to you.

Court ordered Child Support is not being paid by my ex husband.

You should file for enforcement and seek to do so.

What happens when a payor moves out of state and does not pay alimony ?

You can file for enforcement anyway and you probably should do so.

The short answer and unfortunately the unfair answer is no he cannot collect. The time to seek the child support was while she was alive although...

Can I motion to have alimony direct pay as opposed to via Probation. My child support payments are ending in May.

It is not ordinary for alimony to be handled via wage execution.

Is it better...

You should go in person so a judge can see you.

How can I collect child support, as per a divorce decree, from an ex-husband who is no where to be found.

You can ask court permission to use his social security number to find him.

When you subpoena an accountant for your adversaries records, what cost can the accountant actually recoup/charge?

The respondent is entitled to copying and research fees.

Our lawyer did not attach to rape suit, any expert forensic report to support claim for damages. Is this Legal Malpractice?

It may be and it behooves you to speak to a legal malpractice attorney.