What's the fastest way to get a copy of an I-130 approval notice?

It depends who is requesting it and why.

Just go to the Secretary of State, and you can also request them at the Daley Center 10th Floor I believe.

Potential Consequences of being out of status for less than 6 months?

There is no bar to readmission for under 180 days unauthorized presence, however they may wish to consider that at the U.S. consulate if you apply...

Immigration

To get your sworn statement and ask for any evidence you have other than your assertion.

Can two misdemeanors be a base for L-1 denial?

You will certainly be rejected, however you can apply for a nonimmigrant visa waiver.

What happends if a immigation bond is not paid to the cosigner and the immigrate?

Is this a bond for removal proceedings, or public charge bond, or something else?

If the country she is from is a Hague Convention country, very likely you must comply with the Hague Convention before you can adopt the child. If...

Can my mother apply for adjustment of status if she entered the United States illlegally?

If she was admitted, and can prove it, she can adjust. If she entered without inspection, she cannot adjust unless she is 245(i) eligible.

Out of status F1 student

No, you cannot file for adjustment for anything but an immediate relative petition, unless you maintain lawful nonimmigrant status. If you did not...

I am a US Citizen Married to a Thai woman for 5 years. She has a Permanent Resident.

If you filed the I-864, you are obligated for the rest of her life, or until she naturalizes, to maintain her at 125% of the poverty rate.