I have an HP Pavilion, Windows 7, that I haven't updated! lol (and am buying a new computer myself - soon!). Yes, the Western Digital (WD) external hard drive will work with Windows 7, and yep, it just plugs in. So you should be able to put copies of your files on the hard drive and then put copies on your new computer so they'll be in two places. Just make sure a new computer has ports for the external drive to plug into. (and that it takes whatever media cards you use).
I'm getting one from a computer store in the city near me that has options. What I don't like about ones from chain stores like Best Buy etc. is they only have midpriced ones geared for the general public, for gaming, etc. Those would have features I'd never use anyway. I don't really want to spend the money on more memory (16 instead of 8) but am going to, because I know if I don't I'll probably be sorry later! But maybe you wouldn't use that much memory to be worth the cost.
I use Photoshop Elements, just bought it, no subscription (it just seems the subscriptions are promoted more). I don't do much editing, just mostly brightening up esp. for printing, adjusting contrast, that's about it. I'm a longtime film photographer so early on I learned how to frame and compose the way I want and how to get proper exposures. It sounds like you'd be fine with the Picasa you have now or something similar would do the trick.