It is grounds of inadmissibility issue. Since this is not a felony, you should be fine but you have to disclose it.

I-864 country of domicle

Country of Domicile of hers should be in U.S. if you don't want to create confusion to your case. Temporarily living overseas doesn't make the...

Traveling outside USA

Return permit and obligation to file tax return are two totally different things. You should consult your accountant for advice.

Being on a dependent H4 visa, can the person work for a non- profit organization? The position in the non profit is still paid.

If you are paid, you are employed, which is not authorized by law.

Do I have grounds for a Civil Suit?

You may have ground for lawsuit against the other employee. But for $250?

How can I count my unlawful presence days?

USCIS may have a problem. Your attorney will surely find straight forward answer about the accruement of unlawful presence.

How long do I have to live in Utah to be considered a resident for the purposes of changing my name?

You can find rules on Utah's Court's website. http://www.utcourts.gov/index.html

I have rented to own a house for three years and paid a 1000.00 deposit plus 500.00 $ rent and never late on payments.

it's odd that Bank of America can do such a thing. You need to make sure your understanding is correct and on paper. You should collect all papers...

What is the easiest and fastest way to have my fiance with me in the u.s. ?

If you marry him and have DNA proof that the daughter is his, you do have better chance. Of course, the arrest record is not going away...

Can i get in trouble for cashing a check that was gave to me

This depends on what evidence you have to support you position.