Onions should be stored in a well ventilated, dry place. Don't store them in piles; single layers ensure longer life. Old panty hose are an excellent place to keep them. Just slide an onion in and tie a knot. Slide another in and tie a knot etc.

If you have an extra refrigerator, that is an excellent place to store dry onions.

Chopped fresh onions will stay fresh in a screw top jar in the refrigerator for several days.

To Freeze:
Chop onions up in normal cooking size, i.e., diced to 1/4" or less.  Spread diced pieces out on aluminum foil and freeze.. It only takes a few minutes.  Remove from the foil and immediately put in freezer bags. Diced pieces were frozen individually and remain that way in the bags.  They keep well and you can remove from the bag only what you need because they're not clumped together.

To remove onion pungency from the hands, rub well with vinegar, salt or lemon juice. Then wash hands in hot water and soap.