I don't have the facilities to properly cold crash but my attic at this time of year is pretty cold - about 8-10°C (50°F). Would there be any benefit to leaving my beer there, maybe for an extended period, longer than a typical cold crash, or does it have to be really cold to work regardless of time.
It won't hurt anything for sure. It will work about how you'd expect, actually, your beer will clear up quicker than at room temperature but not as fast as if you could get it down to 35 F or wherever you'd cold crash at. Some things like chill haze, you won't get to settle out unless it's cold enough to actually see it.
50 F should be cold enough for gelatin finings if you are concerned w/ clear beer. You could also probably place the fermenter in a container or ice water or a swamp cooler setup to get it a few degrees cooler.