Question about Teufel 3sixty

Yes, that's no problem.

When I click on the Spotify icon, I get this message: "Use your phone, tablet or computer as a remote control for Spotify. Go to spotify.com/connect to learn how"

Unfortunately no; only via smartphone etc.

Yes, but searching on a smartphone is a bit more convenient because you have a "real keyboard" and don't have to select the individual letters with the dial.