It's difficult to answer this question without looking at a copy of your green card. You should seek out the advice of an immigration attorney for a full consultation. Perhaps it's just a matter of applying for a new card if the one that you have has expired.
Is your child a U.S. citizen? I am assume that he/she is. You and your child are different individuals. I will answer generally since I do not know what type of benefits your child is obtaining. Having your child obtain benefits is separate from you obtaining public benefits and this should not necessarily hurt you provided the benefits are issued to your child (and not you). But you should speak to an attorney for guidance on your specific immigration matters.
A K1 requires a prior approved fiance petition (form I-129F) petition for an alien fiance. It takes several months for approval and you will most likely not get it approved by June. After it's approval you can obtain a K1 visa at the US embassy abroad and return to the US as a fiance(e). You should seek out the assistance of an immigration lawyer to guide you through the best choices for you.
Your dad must obtain a consultation from an immigration lawyer to assess whether or not he should apply for any type of immigration benefits. It would be helpful if you could take copies of whatever court papers you may have concerning his conviction. Do not apply for anything without seeking full advice from a lawyer.
If the U.S. state department has issued you a United States passport then you are an American citizen. A passport is definite proof of U.S. citizenship.
A consultation with an attorney will review factors such as your immigration history, how you entered the U.S., if there are any crimes involved or other outstanding issues that may eventually effect your case, etc.
I would need more information about you to determine your eligibility for a green card as there are many factors involved. You should therefore obtain a detailed consultation with a lawyer. Generally speaking though if you are here on a visa and you are married to a U.S. citizen your American spouse files a petition for alien relative for you and you file an application to register your permanent residency. You can also apply for a work permit along with these applications.
You will need to file an I-130 petition for alien relative for your husband. It will first get approved by the USCIS and then the file will be forwarded to the National Visa Center and then the U.S. consulate. He will then have an interview and obtain an immigrant visa stamp in his passport. Afterwards, he enters the U.S. and is approved for permanent residency at the port of arrival. This should take about 6months or so. After you file the I-130 you also have the option of having him come...
It's difficult to answer your questions without looking at the entire petition and the request for evidence. I do not know if your employer filed the visa extension petition on their own or used a lawyer. You may want to seek out legal advice from an attorney and have the attorney review the job description and your specific situation carefully including details about your job and the employer. There are several ways of proving that an H-1B requires a bachelor's degree. You need someone...
This is a form that is filed by your employer with the U.S. Department of Labor. It basically indicates to the labor department that the employer has met certain requirements for your H1B visa. There are several attestations (i.e. there is no strike or work stoppage at the company). The form will indicate your rate of pay and the prevailing wage for your job in your new geographical location. It was probably filed as you changed locations.