Rehab center admissions paperwork. Do we HAVE to sign?

If you do end up signing the admission papers, please be aware that many include a page that declares that you agree not to sue them and instead will settle any disputes through arbitration. If you sign it, you are signing away your rights to sue them in the future. You are not required to sign that if you don't want to.

When my mom entered rehab, I was asked to go to the facility the day before my mom started to read and sign all of the paperwork.

As GA says. They need paperwork done to bill Medicare. He used 10 of the 20 days that Medicare pays 100%. If you are POA you sign the paperwork with POA after your name and sign nothing that you are responsible for payment.

Be aware that in the new place Dad only has 10 days at 100%. He does not start over again with the first 20 days. 21st to 100 days Medicare only pays 50%. Dad will be responsible for the other 50% unless his secondary pays the balance or partial.

I suspect the staff wants the admissions papers signed so they can get reimbursement from Medicare.  If someone doesn't sign, they may and/or probably will bill your father directly.   What I would do is carefully review the papers, make changes as you want, copy the whole set and then return them, in person if possible, to the first rehab facility.    If they give you flack, you might mention that their services didn't meet standards, and if they continue to harass you, you'll be forced to file a complain against them for lower standards of service.

Make a list of the conditions he developed while there and use it as leverage.