My reason was stupid . Should I tell them true or?

J-1 usually allows for a 30-day grace period, but you must be truthful about any previous stays in the US.

Should I answer yes to this question?

As long as it was actually adjudicated and not rejected solely due to the lottery, then YES.

Will I be eligible for a Green Card from an E2 visa?

In very general terms, it seems that you have a fine chance for a successful sponsorship. Consult with an attorney to explore the details.

Can one travel for H-1B stamping while H-1B amendment (new title and salary) is under review?

Yes, you should wait for the amendment to be approved before traveling to minimize any confusion.

The EAD is available only if you and your spouse are maintaining H-1B/H-4 status, which is independent of the I-140 petition.

Can L1A holder file for I-539 for extending stay beyond max stay limit

No, you cannot remain in the US in L-1A status beyond the 7th year by filing an I-539.

VAWA is approved, should I request Dallas immg.office to transfer case to vermont office or NY city office, where I was moved?

The case will be consolidated in Vermont and may then be transferred to NY where you reside for any interview.

Which one will be processed faster? F2B or EB2 or F1?

Subject to change, the F-2B category appears to be the fastest route as of March 2016.

If you are earning money for work performed in the US, it would be a violation of status.

She must prove that you are married and that you are in fact maintaining valid H-1B status.