When I do strenuous activities outside, is it normal to get sweaty *long* before I get any sort of physical exhaustion?

Sure. Sweating is your body's way of keeping you cool and has nothing to do with the amount of effort being exerted. People can stand still and sweat like crazy. People can REALLY exert themselves, but perhaps not sweat much at all. 

Yes, it is normal. Sweating does not mean you are physically exhausted. Sweating is a means by which the body cools off. When you are physically active, your body is producing excess heat. That heat has to be transfer outside of the body else you would collapse from heat exhaustion (or heat stroke) which can kill you.

You should never exert yourself,  especially not to exhaustion.   If this is pre-bedtime, light exercise means a short walk or 15-20min slow on the elliptical.