Unlike the constitutionally protected right to remain silent or the right to representation by an attorney, you do not have a right to refuse to submit to fingerprinting once charged with a criminal offense. I strongly encourage you to take advantage of your other rights by posting no additional statements on a public site (including Avvo, Facebook or Twitter) and retaining an experienced criminal defense attorney to represent you in this matter. Good luck.
You are required to be fingerprinted. You should get that done at your earliest convenience. More importantly, you should hire an attorney to dispute the charges.
The above is a general answer but should not be considered specific legal advice for your case. It is in your best interest to contact a qualified attorney to discuss the particular facts of your individual situation.
That varies from state to state, in many states, the law allows it, that doesn't mean the police do it every time for minor arrests, but they have the right to