Whether you have derived citizenship depends on a number of factors including your place of birth, your father's immigration status at the time of birth, since when you are an LPR and so on. Your question is rather nebulous on all of these issues. Talk to an immigration attorney here in New York.
You should get a receipt and later request for additional evidence will most likely come as well. You will need to read it carefully and provide USCIS with the information they will ask. You are allowed to stay in the country as a tourist for the duration of adjudication.
It appears to be a CIMT and depending on the time you actually spent in jail, it might or might not be a problem. It might also be considered as domestic violence and then it will be a problem. It also can raise discretionary objection.