Legally what power does your wife have to move on with a separation and divorce if you refuse to sign the papers?

If most states, you do not need to prove "fault" to get a divorce or legal separation. She can hire an attorney and proceed however she deems fit.

Answer

She has every right to proceed with the divorce. She does not have to have your permission to get a divorce.

Answer

She does not need your signature. If you refuse to sign to acknowledge that you received the petition/complaint, she will have you served. If you refuse to sign a settlement agreement, she will ask for a trial date and have the judge decide how to divide your property and to declare you divorced.