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Well, looks like you have a filament checker, and kykes, depending on the characteristics of the particular tube (filamen voltage) the light may be feeble or bright. You can do the same thing if you have a $10 digital volt-ohmmeter by checking the appropriate filament pins for continuity.

Tube information here for just about anything you are likely to have:
http://www.nj7p.info/Tube/SQL/Tube.php. Pin diagrams are shown from the bottom of the tube and numbers increase anti-clockwise from pin or gap locater.

Nothing else (shorts, emission, even lack of vacuum) will be known by checking filament continuity, but if the filament (or heater is open it is a dud.

Tube sockets can sometimes be cleaned up and carefully retentioned so as to make good contact with tubes. Likewise tube pins can be cleaned off and scraped if need be. Careful clening tubes, sometimes the identifying number is a decal which can wash away. So proceed methodically, and re-label tubes with a bit of tape if need be.

As for a loose base, it is sometimes possible to glue it back together (I use Elmer's, let it soak in for a minute, invert tube in a can or whatever, leave for a week, scrape any glue runs that seep back down in the glass, and test with a reliable full function tube tester before even thinking about using in a set.)

Hope this helps.