is it up to me to cancel the direct debit with a previous energy supplier after i have switched energy suppliers or ...?

Yes, close that account.

Yes. A direct debit authorisation on YOUR account is what YOU signed, so it will stay there unless you tell the bank to cancel it. No energy supplier can cancel it for you. It's best to make sure you have paid everything you owe to the previous supplier, THEN cancel the direct debit so they can't charge you again.

yes

It's up to you to start and stop services.

Usually your new supplier does it for you, but you should check your bank statements thouroughly the next three months, because sometimes they forget. I wouldn't do it myself because, if you're bad on luck, that interferes with your new supplier's cancellation, and you either have a double contract, which will be expensive, or none, which will be bad.