My J1 visa is expired on 08/31/2013. Do I have grace period?

You have a grace period. Do not leave the US or you probably won't get back in. Here's a link to brief explanation:...

Can getting married before your filing is finished stop my immigrant visa processing?

Do you mean that you (or techically your fiance) already applied for a fiance visa (k-1)? If that is the case, then this could be an issue.

How do i maintain my status in usa before shifting to H1b from oct 1? I am currently on l1 a from Comp A.

Call Company B's immigration lawyer. You could get a B2 to fill in the interim but these are rarely granted.

I desperately need help I took the citizenship test and past all the parts but written I try but just can't get it

There's not a good answer here. You need to be able to write. There are prep courses, but those don't really help if there are other issues like...

Does this impact the postings? There's a good chance that's your answer. Presumably you already have a lawyer handling this on behalf the company;...

US citizen and Want to petition for my mother's green card

you need other evidence to corroborate what you're saying. the passport is a good start. maybe other records with both of your names on it, if any...

Convicted of aggravated felony, married to U.S. citizen, 4 children, I-130 approved, what now?

You need to work closely with an attorney to see if there are any waivers you can take advantage of. However, I do not see this as terribly likely....

I'm a physician and want to continue my postgraduate training in US but I'M AGED-OUT? What should I do? Please help :'(

Why was the J denied? An H might be an option. There's a lot going on here and you need fact-specific advice. You may want to speak about this...

I had filed for my Green Card in July of 2007. I received my EAD in December 2011and since them we have been renewing our EAD.

You could try to do an infopass. Are you working with a lawyer? You should call him/her for guidance if you are.

Can an I130 petition be removed by sponsor?

It can be withdrawn, but you may have other options in that scenario. You should call your immigration lawyer ASAP.