The sun does not act as a catalyst during evaporation.
Evaporation is not dependent on sunlight.
Acetone is a simple hydrocarbon that forms water and carbon dioxide when burnt.
While taking precautionary measures and wearing suitable protective clothing is a good idea, worrying about wearing shorts while handling small quantities of acetone is overkill. Acetone is only slightly toxic and generally recognized as safe by the FDA.
You won’t. Acetone is naturally found in the human body and is not particularly toxic (as discussed above). I would not recommend pouring acetone in your eyes, but it is not known to cause blindness. If you do get a little in your eyes you should 1) feel great shame in knowing that you can’t handle a liquid without getting it all over your face 2) flush your eyes out with water and 3) call it a day.
The primary irritation of the skin caused by acetone is dry skin. Don’t bathe in it and you will be fine.