Yes it does not matter who pays the fee.

The valid AP should be sufficient for reentry.

You may submit an expedite request if possible.

File the renewal as soon as possible.

You may file a motion to reopen or refile the entire application.

That is for fingerprinting and background check.

Yes you may do so or wait for oath date.

Depends on the basis for refusal. Consult with an attorney for guidance.

Best to apply for N600 naturalization certificate.

Best to consult with an attorney for further guidance.