I was caught by the Police driving without supervision on a provisional licence. What penalty do I face?

The normal allegation would be driving “otherwise in accordance with your licene” (offence code LC20) which is punishable with 3-6 penalty points and a fine. However, this offence would also mean that any insurance in place would be automatically void. Consequently, the Police will probably prosecute for that offence as well which carries 6-8 points and a fine. Individually, or combined, these offences could result in a discretionary disqualification.

I was banned for 12 months for dangerous driving. Do I have to apply to the DVLA for a provisional licence?

Your licence will have been revoked by this punishment.  As a result, you need to apply for a new driving licence.  As you were also ordered to re-take a driving test, you should apply for a provisional licence, but indicate to the DVLA that you need to take the extended test.