Steam – see cards for games I do not own

You can craft badges for games you don't own, yes. You'll need to trade for them or use the marketplace, but it's definitely possible. You can also open boosters, but you'll similarly need to trade for/buy them, since they only drop when you have received all your normal card drops for that game, but you can't get the normal drops without playing it, which requires you to own it.

No, you cannot see if a game has Trading Card support from its Library page. If you want to see which of your games support Steam Trading cards, use the Badges section of your Profile that you mentioned. If you want to check a specific game you own, you're gonna have to look it up there, or just search around a bit.