Put the name she is currently using. In many states you don't need any additional document for that - a marriage certificate is enough,

I'm a USA citizen , my sister is here visiting me, I know I can file for my sister to come live in the USA ,

To adjust status one must be in status unless adjusting on an immediate relative's petition (in her case it would be a US citizen husband). I...

I applied for i601 back in June 2013 I haven't hear anything yet and it's already been over 9 months what should I do

Have you received notice of receipt? If yes you are still within the normal range. If not you misfiled. In either case see a good lawyer.

It may take several months as they experience backlogs now. .

Can a US-based company expedite the immigrant visa process for a Canadian citizen once the I-130 has been approved?

TN visa does not allow dual intent. Therefore be careful not to have the fraud issue involved in your case.

Could I be eligible for a L1B visa with an immigrant (family) visa petition pending?

L-1 allows dual intent. Processing can take very little time if you use expedited (premium) processing - literally two weeks. .

Selected as the best answer

No big deal at all. Use whichever name but also mention the other when asked if you have used any other names.

I'm a convicted felon and been in the u.S since I was 3years old I am Now 25 and my green card expires next year...

the answer to your question must involve an in-depth analysis of all the facts, including the criminal records (certificate of disposition, plea...

Do I need Form G-325A in order to sponsor someone?

No, this form needs to be filled out by the beneficiary only and by both the beneficiary and the petitioner when it's a spousal petition.

I751 wavier denied for fraud; i filed my wavier but denial notice just arrived .

Your status will be "in removal proceeding". As the courts backlog is huge you may have to wait well over a year, in fact close to two years.