The derrailleurs you mentioned are all compatible with a 7 speed system. The shifter controls the amount of cable pull and not the derrailleur. As for the M580 this is called the "Rapid Rise" derrailluer? If so you won't find much good reported about them. And the fact that they were dropped after only a year or two should indicate something. So best to stick with regular pull models. If you want to upgrade to 8 or 9 speed then just change the cassette and shifter along with the derrailleur. And obviously if you go with a 9 speed upgrade then you'll want a 9 speed chain.

I had to look up "shimano shadow". Looks to me from the google hits that this is a new variation on the XTR level. It's up to you if you think you need to pay that much more for something above the very capable LX or XT grade. I know that I've always been happy with LX or XT. I've had one XTR rear derrailleur on all my years. It worked well but I sure didn't abuse it the way it was used and it didn't work any BETTER than my LX or XT stuff.

For chains the 7 and 8 speed stuff all works with the 8 speed chains. But 9 speed needs a different chain that is a little more narrow.

I'm an enthusiastic SRAM fanboy. I like the Powerlink's ability to easily take the chain off and put it back on without needing to push and replace pins. But any of the chains you mentioned will certainly work just fine.