Hink-Pink

Gnats' hats.

Wet pet is the hink pink that works for both cats and dogs. When you give a dog a bath in Hinky Pinky, the end result will be a soggy doggy. When a cat gets a bath in Hinky Pinky, the end result might be a mad cat because cats do not like water.

Mouse House.
Mouse house, or rat flat.
If you mean what is a hink pink for a home for a rodent, it is "rat flat" or "mouse house".

Best nest is the hink pink for superior bird home.

damp lamp

funny bunny

bear scare

banana havana

a sick chick?

stout trout, overweight skate

hairy berry, hairy cherry

Lazy daisy.

Wild child.

unclean queen

Rounder Flounder.

Witty kitty

Shy fly.

sick tick, infirm worm

rear dear

Slick chick.

What is hink pink of a rhyming pairs that stinging insect that doesn't cost money

Grape ape.

clear steer

It is fake snake.

Fair hare.