1) The tape stuck to the tyres is coming loose along the sides. Can I
re-attach it? If so, what kind of glue should I use?
Yes. I've always used ordinary rubber solution - layer on both tape and tub, leave to dry, then stick together. It's always amazed me that it's adequate, but it is. Others use Copydex.
2) How clean do I need to get the rims before sticking the tubes back
on? I've got rid of most of the old glue, but there's still a bit round the
spoke eyelets. Do the rims need to be really really clean?
It sounds as though you're fine with what you've done. As long as there's no huge build up of old cement you'll be fine. I've hardly ever cleaned rims - the tubs usually take enough of it off themselves, especially with (the very thin) modern cement. You certainly don't want to go struggling to get the lot off!