Is Pti programm my best option?

This question cannot be answered in this forum. More information is necessary. You need to consult with an immigration lawyer, preferably one who...

Expired green card?

You can apply for naturalization with an expired green card. YOu may also qualify for a fee waiver.

Can I visit my country if I have investigation against me in the us and I only have a green card?

Much more information is needed. You need to consult with a lawyer, preferably one who specializes in both immigration and criminal defense.

Since may 26 2014 i have a greencard from lottary , i have been outside united state more than 6 months but less then 1 year

You have got to be kidding me. You need to consult with a lawyer.

I need to know the physical presence and continuous residency exceptions and what can be done to avoid these conditions

There are no exceptions to those requirements.

Can I travel when I get married?

It takes about six months to get the travel document. I strongly suggest you consult with a lawyer before filing anything. The process has more...

Will I be able to renew my green card and or apply for citizenship?

Maybe. More information is needed. Start by consulting with a lawyer.

I-130 Form, Do I enter "Other Names" ?

You need to review this and the documentation with a lawyer. This is too specific for advice in this forum, and the process is more complicated...

It is possible. All the evidence needs to be reviewed. They could have taken blood or urine. Hire a lawyer.

Will using NJ Family Care (Medicaid) and Housing 8 Voucher impact the green card renewal?

No, that should not affect their renewal. If they speak English, they can naturalize. IF not, they will have to wait until they have had their...