Oil

You can,but i think your car use,s 20w50,10w40 is heavy oil,

and use syn.oilsIf you want to put 5w-20 in there, then feel free to do it. Honda actually reccommended 5w-30 in your engine when it was new, then several years down the line, they changed the reccomended viscosity to 5w-20. 5w-20 is a little thinner so it will give you better gas mileage (barely noticable). There is no harm in switching right now.

Yes

If you do want to use synthetic oil, first check the entire engine for any signs of oil leaks, and have them resealed. Synthetic is way more expensive than regular oil so you wouldn't want to be wasting it through oil leaks.