Am i eligible for Visa waiver program, I came to USA on L1 visa in 2010 and moved on to H1 visa in 2011.

You have to be a citizen of a country which US has a visa waiver program for.

Will all of the dirty awful details of my messy divorce be public information to prospective employers or other courts?

They may unfortunately so to be better prepared I would recommend discussing it with the workers comp attorney.

Can an immigrant married to a US. citizen apply for divorce under US law?

The immigrant may be able to - it's best to consult with a divorce/family law attorney!

About L2 visa

If your L1 visa is still valid your wife and kid can get L2 visas. I would recommend speaking to an attorney about the salary you will be receiving...

When to apply for State ID when applying for i-130 ( Immediate Relative of US Citizen )

You need to wait for either employment authorization document or if you are not applying for one - for the actual green card.

Working on H1 receipt, related concerns

You can start working based on H1B receipts but if denied you will be the out of status from denial date.

How can i look for a job in USA legally? can i go with my tourist visa and look for a job?

Yes, you can look for a job now, but need to get employment authorization to work legally.

State ID/DL during i-130

Neither. Need to either have employment authorization document or Green card for the ID/DL.

Urgent H1B Matter - need help with RFE (Request for Evidence). Seeking second opinion.

Sorry to hear that you are in this position. How off is your annual salary from the requirement? For your situation I would recommend scheduling a...

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Travel on the Visit visa the one you got before you married and don't have an intention to live permanently at the moment in US.

She can travel on tourist visa as long as she is not planning to stay at the time she enters the U.S.