Wouldn't it be more appropriate to say "a sheet of paper" instead of "a piece of paper"?

What's "checkout"? Is it only used in grocery store?

A "checkout" is the location in any store where one can purchase goods from the store. Ex: The cashier works at the checkout. Ex: He went to the checkout to buy headphones.

When do you usually say "geez"?

Great question! "Geez" actually comes from "Jesus". It is an informal way to express surprise, disappointment, frustration, annoyance or other strong emotions. "Geez" is an example of a euphemism (a word or expression used in place of one that is considered offensive or suggests something unpleasant). Using the name of Jesus Christ to express a strong emotion unrelated to spirituality was considered blasphemous, so "geez" can be used instead and became quite popular. Ex: Geez, you don't have to scream if you disagree with me. Ex: Can you believe the way she acted at the party last night? Geez.

What does over-heating mean here?

"Over-heating" here refers to your body getting too hot to the point of being unhealthy. Some ways your body could over-heat is by exercising hard, being in a hot environment, etc.

Is "she's" an abbreviation of "she has"?

Yes, "she's" means "she has" here.

What does "tank" mean here?

"Tank" here means "aquarium", which is a rectangular glass box that fish, amphibians, and reptiles live in when kept as pets (or to eat). Ex: My pet lizard lives in a tank in my bedroom. Ex: The fish market had tanks of fish.