To start a creature, black and white;
A PANDA is a black and white creature.
In myth, I may give you a fright.
PAN was a Greek god whose appearance caused terror. He lends his name to the word panic.
I may be found on top a flame;
A PAN is a cooking utensil, often found on stovetops.
Eternal youth when I'm a name.
PETER PAN never grows up.
5 rings, but only in the West;
OP revealed the intended answer is the PAN AMERICAN GAMES, which resemble the Olympics, except that only countries from North and South America compete.
Reversed, I give a little rest.
A NAP is a short rest. (This was my inroad to the solution.)
A box or jewel in your ear;
PANDORA'S BOX is a famous artifact from mythology; PANDORA is a popular music streaming site; PANDORA is also a brand of jewelry.
To beg for coins from strangers near.
To PANHANDLE is to beg for money in the street.
Connecting bodies that are wet;
The PANAMA CANAL connects the Atlantic Ocean with the Pacific Ocean.
Add a letter, I'll heave and sweat.
To PANT is to breathe heavily, often while heaving and sweating.