Can you craft badges or use card boosters for games you don't 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.

You CAN:

• Craft badges for games you don't own
• Open Boosters for games you don't own

You CANNOT:

• Craft Boosters for games you don't own