There is not enough facts to advise you properly here. And you may have a case for the domestic violence or VAWA. Also, if you are in removal proceedings, then you should have a lawyer representing you and you should consult with your lawyer about your case. If you don't have an immigration lawyer, I highly recommend that you hire one, because your case is very serious.
Follow Gintare Grigaite, Esq. on Facebook (search for Gintare Grigaite, Esq.). Contact immigration lawyer Gintare Grigaite, Esq. of Grigaite & Abdelsayed, LLC at 201-471-7989, located in New York and New Jersey, for a consultation about your immigration case. Answers on AVVO do not constitute legal advice and do not form attorney-client relationship. Always consult an attorney for a legal advice.
Yes, failure to pay child support is deemed bad moral character.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.