1.  The only thing I've ever used, or seen anyone use, a sink aspirator for is as a vacuum source for aqueous and otherwise non-hazardous (or regulated) materials filtrations (i.e., mostly biology folk).  Otherwise, the use would be inappropriate for all the reasons mentioned by others.

2.  The type of trap you would use depends on what you're doing.  A liquid nitrogen trap is fine for schlenk line work, where you're not really pulling air through something, just more evacuating a small volume.  If you were to use liquid nitrogen for say a filtration, that would be bad because you could condense liquid oxygen.  Dry ice/acetone would be a better option in the case.  If things are aqueous, and otherwise non-volatile and non-hazardous, you can just use a non-cooled trap to protect your vacuum source in the event of overflow.

3.  House vac is fine for filtrations.  For rotovapping, you would likely need a pump for anything with a moderately high boiling point.  The diaphragm pumps are the norm these days for such things, but leave a lot to be desired in terms of performance and robustness.  They're also quite expensive to fix when they do (and they will) break.  For schlenk line work, belt driven pumps are dandy.