Can i apply for Eb2 cat green card if im currenty on a F1 status?

You could, but it is best that you change your status to a H-1B and maintain the status until you become a permanent resident.

Can I travel out of United States when my current status in on H4 and COS to H1 visa under cap (2013 Oct) filed and approved

You can travel back to the U.S. with your valid H-4 prior to the 10/1 H1 start date.

What r the documents required to travel and enter US with L2 Visa(dependent)

Your family need valid L2 visa.

HI I am on H4 visa and will apply for COS from H4 to F1 on 20th June. Can anyone please tell me the processing time for COS?

Around 3 months.

Husband under NIW-EB2, can I apply?

Unfortunately no. In my experience, USCIS will take 3-6 months to process your case.

Will I get I-94 issued on an H1 extension with future start date of extension while I am in US on H1 currently?

You have to leave the U.S. after 7/29.

What should I do If I am laid off on Cap gap opt period ? and my H1B will come into effect from oct 1 ?

As a F-1 OPT, you may have up to a total of 60 days of grace period after your OPT expires. You can continue to stay in the U.S. during the grace...

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Can I reuse remaining ~4yrs of H1B which was approved in Oct 2004 ?

Your employer has to file for a new H-1B petition.

Does overstay your visa in Usa make you banned also from Canada

Probably no, but you need to talk to an attorney who is specialized in the Canadian immigration law.

Will my EAD still be valid once I switch employers via H1-B transfer ?

How did you get your EAD? Is your EAD from your prior status, or you have a pending adjustment status application?