Yep on ^^^^ as well as play all notes. Hear for any out of tune or muted or dead notes. They do exist on certain guitars as a phenomenon of sorts.
Try the controls, fast and slowly to see if they fit well for your playing as well if you like the tone roll off or volume swelling/gain control on the guitar. Check the electronics at every angle, their your guitar's 'engine'. Check the input jack, pickup selection is clean and snappy and that all options work.
Play your favorite stuff to see if its a good guitar for that. Say you like jazz and want a Tele, go for the neck or middle position, roll of some of the tone knob and see if the sounds coming out are pleasing. Which they should be
Or if your gonna do some modern metal, hit that bridge pickup and play it as you would Live at Donington. Donington is still considered modern?
Bend the high notes high up the neck and check for fretting out, meaning notes going out of tune or being muted because of uneven fretwork.
Of course see the neck for any bowing or bad relief. Hold the guitar like aiming a rifle and focus your eyes down at the area where the Estrings and the neck are. You can see if the neck is curved up or back or straight.
Condition is up to the individual, of course if its new you would want it unblemished. If the cosmetics are a bit worn, do ask for a discount or ask for another piece, and start everything all over again. Don't be unreasonable, you think they would give you a 300 buck cut on a 750 quid korean copy?