I was in the same situation some years back. In this case, i was allowed make AVC’s until the 6 months had elapsed, but, scheme rules can differ.
Yes, is the answer to question 1), allthough if there is not enough pay in your December payslip, which might well be the case, you can make a payment to the provider, post December, and make a tax claim next year, and recoup the 40 % tax relief next year, there is a deadline of October 31st 2024 to make an AVC tax claim in relation to 2023 tax year.
2. It is restricted to the current employment earnings, eg 20% of the Gross Taxable income for employment of your new employment.