My bf has an Elgato, and it's great! There are a few things it's finicky about, but for the most part it works great.
This page should help you with what the capture card is compatible with and how to make it work:
http://help.elgato.com/customer/portal/articles/1020266-what-game-consoles-can-work-with-elgato-game-capture-hd-From what I saw in there, you should be able to capture NES titles just fine.
Even with that list, don't assume that you can't stream a certain device because it's not on there. I've hunted around on the internet and found that I could stream my PSP-3000 through the Elgato as well with the correct cables (component).
Also, if you want to stream PC games, no need to have a capture card! You can just use OBS (
https://obsproject.com/) which is free and fairly easy to use. The hardest part is probably the settings, and that's going to be different for each person and their system specs, but usually Twitch folks are really nice about helping answer those questions when you ask. The Twitch site itself also has a page on how to setup your OBS for streaming.
If anything, for PC streaming, your rig should be of a decent quality - my old home-built computer of about 4 years ago was decent for playing Steam games (usually at low-to-mid settings) but couldn't handle having the game up and OBS at the same time. OBS isn't that CPU-intensive - my computer was just not up to specs to be able to handle both programs simultaneously.
