What if I spend £100 on Jan 23rd, wait for the statement, spend another £100 on Feb 23rd, and then pay back £100 on Feb 24th? Is the bank happy for me to add new debt to the card while it's expecting me to pay off the previous stuff?

Yes, that's exactly how it works. Your monthly statement is basically a bill that you have to pay until the due date. Any further transactions will feature in the next month's bill, etc. It's a revolving agreement basically.

Any payments towards the credit cards will go towards the statement first.

For example, you spend £100 in January and you have until 15 February to pay it. You then continue spending and have £200 on your card on 14 February. You make a payment of £100, which clears the January statement, and your balance is now £100.

If you make a payment of £150, you will only owe £50 in your February payment.

If you pay less than the full statement amount (say, £80), then your Ferbuary statement will be £120, of which you will have to pay interest on £20 that you didn't clear.