Can my h1 797a be approved again?

If you had filed for and were granted a change of status and you never went to work you may have violated your H-1B status. If you did not change...

Specialist physician - Hardship waiver approved

Yes you may have tow applications for resident status pending concurrently. Speak directly with an immigration attorney.

generally speeding in not by itself a character issue.

This is a tax question not an immigration questions as my colleagues have indicated.

H1B Amendment

Speak directly with your employers immigration attorney.

Do I have to apply green card again for my children when they complete 14 years?

The I 90 is the correct form and the I 90 needs to be submitted whenever the cards expire.

I was recently married on the 7th of December 2014 and got the marriage certificate on the 8th.

I concur with my colleagues this is not an immigration question please repost.

I agree with my colleagues. Speak with an immigration as this is not a status but there may be something for her that is in the INA.

For Denied H1B , A Fresh H1B becomes cap exempt?

Denial does not include you in the CAP.

What happens if I apply for the Green Card (married to a U.S. Citizen), while my F-1 OPT application is still being processed?

Yes your wife may petition for you and you may file for adjustment if eligible while the OPT is pending. Speak directly with an immigration...