Will my 10 year B1 visa be active if L1B is issued during B1 validity?

You can travel on any valid visa on your passport as long as it is in line with the purpose of your visit and it is not cancelled by the CO.

I am a LPR of US. If I apply for unemployment, Immigration can refuse my application for citizenship?

You should be fine with your employer's insurance provider.

My H4 to F1 COS got approved and F1 visa is valid from December 2013 . Can I travel to India and come back before Dec on H4 ?

Not recommended.

No. Usually this will come up at a Consulate abroad when you apply for a visa and the CO will determine whether you failed to maintain status.

I am with L2 EAD as Full-Time employee for US based company. Can I work from India for the same company (No branch in India)?

This is not necessarily an immigration question.

I filed I-751in NOV,2012. Got Notice of letter for interview.

Follow up with the Department if you do not hear from them in about 30-60 days.

File for green card within US or in India

I'd recommend a full consultation with a good attorney before you file any petition.

I filed I-751 . Got notice of letter of letter for interview. They are asking for more evidence that is joint bank account and o

Are you divorced and filing with a waiver or it was mutually filed and now you separated after the domestic violence incident?

Can i work on i-94 and EAD on L2 if my spouse goes back to India? Both our visas have expired.

No, you have derivative status.

Travel restrictions while H1B denial is under Motion to Reconsider

1) Depends. Reconsideration by Department may take 30-90 days or longer 2) Motion filing will not give you any additional immigration benefits...