What forms should I fill as the intending immigrant? I-864 or I-864a

I-864a if you have worked lawfully in the U.S.

Can I apply for work authorization just a few days I submit the asylum to USCIS?

You can only apply for work authorization after your asylum application has been pending for more than 180 days.

Can a spouse work on L2 EAD and stay in USA until I-94 expiry date is reached, while L1 of primary applicant is cancelled.

You status is dependent on your spouse's status so that you cannot remain in the U.S. and work once it terminates.

Applying for family-based green card while employment-based green card petition is on process

Your employment based applications make no difference. You just file I-130 and I-485 concurrently in your situation.

Can my boyfriend get his papers after marrying me. He is from Argentina,,,he came here 12 years ago on a visa and stayed.

Your boyfriend can apply for a green card as your spouse if you are a U.S. citizen and he entered the US lawfully. Income requirements can always...

You should report them on your N-400 and make sure you paid the tickets.

Criminal charge and Immigration consequences

I do not see an immigration question.

I have an E2 visa, which is going to expire this year if I will not be living in the USA do I need to renew it?

You do not renew the visa unless you want to return to work for the business again. On the other hand, entering without an E visa may be...

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Can I move while I am in removal proceedings?

You can move but need to always update the court about your address. You will still have to appear at the court where you case is currently pending...

Do I need an immigration lawyer?

It is a possibility that you will face removal proceedings. It would be prudent to at least consult with an immigration attorney for a assessment...