This does not sound permissible unless you have a company outside the USA which will be doing the billing.

What is reason that VSC has not process any single application since June 2015?

visa quotas and substantial delays.

Renewing green card and doing a name change?

It really makes no difference when you do it as long as you have documents showing the new name.

What happens if my current EAD expires while I am working and in processing for a new one?

Unfortunately the regulations say that you cannot work until your EAD application has been approved. if your pending case is an I-130 petition...

Should I apply for an Adjustment of status of place of birth on my application? Are there any repercussions or downsides?

I would list Libya as the place of birth and provide the explanation and documents.

Can I hire students on a student visa for unpaid internship?

I agree with Elizabeth - one cannot intern for free without authorization.

What is the best visa in our situation to obtain? During the process would she be allowed to leave the country?

There are several options which you would want to consider such as K-1 fiance visas or filing directly for a marriage-based green cards. These...

Advance parole - I-131

unless it's a true emergency, they wont expedite.

Can he apply for US citizenship with these documents/ Thank you in advance.

He should be ok to file for his N-400 as long as he meets the other qulaifications.

If I go ahead with him, can he sponsor me, and what is the process ? Thanks

He may very well can petition for you but the process may take at least 18 months and you will need to maintain status while you are waiting.