Is it true that spiders and bots don't like document-root or relative paths? I read it somewhere just now; and I don't know whether "don't like" means a spider will act as if there's nothing there and prompty leave; or feel somewhat offended, but compensate and carry on...

Do spiders go in circles? If all my pages have at least one "go back up one level" and lots of "go down some levels", are the bots clever enough to remember where they've been? If they're not... can I tell a spider to ignore just some links on a page?

They don't mind relative paths at all. That's a myth and a poor one at that. It came about because Google recommends using full URLs in their webmaster tips and newbies instantly assumed relative URLs were bad. It's simply untrue.

You should use full URLs if you want to be 100% sure the search engines can follow every link (the chance for an error is .0000001% otherwise but some people like to be sure). It's also good in case someone scrapes your site. They usually screw it up and leave the full links up on their site. That then delivers traffic to your site and helps you find the scrapers.