We have a couple of dead trees in the garden (ivy killed them). Does the wood burn ok seeing as its really dry?

Yup burns fine just make sure it is dry as they hold moisture for along time.

Yes

Standing deadwood has produced some of the driest logs I’ve burnt.

It will be well seasoned, I’m sat in front of a deadwood fire now. No need to lay it up for a year or two, just dry it out if needed.