Is it safe/advisable to board a small single- or twin-engined aircraft while
the prop(s) and engine(s) are turning? I'm just wondering if this is feasible
if you just have someone getting on or getting off (with the pilot being in
the aircraft the whole time).

Advisable? No.
Safe? Depends on too many factors to tell say outright... Front
engined aircraft that boards over the trailing edge of the wing?
Probably safe (although I'd imagine getting some doors open under even
an idle prop-stream would be a pain/might risk damage to the door...
light aircraft doors tend to be very flimsy) Twin which boards over
the wing? Not a chance... WAY to many other variables (how bright is
it (how visible is the prop disc), how much idle blast does the prop
throw?
Done anyways? Sure.

On a related note, how long can you safely leave an aircraft (engines off)
with just the parking brake set,

Kinda a nonsensical question, how long until the next stiff wind
blows? what angle will it blow at? In all honesty, most chalks will
probably do no better than the wheel brakes alone at protecting from a
stiff gust... t takes quite a bit of wind to overcome the static
friction of the wheels. If the wind is strong enough to overpower a
brake, a chalk probably won't do much better- what you'll need is good
heavy chains, or better yet, a hangar.

The flip-side is that in general, if your paying the MX bills, its
really not a good idea to leave the brake engaged, this is simply a
wear issue (Its better to store a hydrolic system unpressurized then
pressurized, If you have the choice). Wheel chalks are slightly more
reliable, and much cheaper, then a parking break, but no
"safer" (better at keeping an aircraft stationary)

and when do you normally put chocks under the wheels?

when park your plane.... duh?