Learn English – Does “peer down on somebody” include the meaning of contempt

to shut something down simply means to make something nonoperational. When you shut your computer down, you bring it into a nonoperational state. If authorities shut a business down (by the way, another term that's used to refer to a company or business in English is operation), they legally close it down thereby making it nonoperational.

This idiom can also have a third meaning:

3 [transitive] informal to stop someone from doing something, especially to stop a player from having the freedom to move around or play well

So, to sum things up, in simple terms shut down just means to prevent something from keeping doing its thing by, possibly, liquidating it.

Example of a more colloquial usage:

He tried to ask her out, but she shut him down completely.

This is how this expression can be used in sports:

Our team will need to shut down their passing game if we are to succeed.

The other definition of "to go down on somebody" you found is totally unrelated.

Basically, when prices go down, things are becoming cheaper. The numbers start out big (e.g, $50) and they end up small (e.g. $10), so they are going down.

For example, someone might say "That game costs $100, but I only have $50. I guess I will have to wait until the price goes down before I can buy it."

In your case, person A is asking whether the TV can be sold for any less money. "On the price" means "concerning the price", as in "We need more discussion on the matter."

Person B might respond "Sorry, that's as low as it can go", without giving any discount, or he might respond "Tell you what, I'll sell it to you for only $2000".