Difference – Difference Between ‘My Fridge Broke’ and ‘My Fridge Broke Down’

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.

Both of those verbs are correct and natural in your example, but the verbs "call" and "call in" are not the same.

"Call" means to make a phone call, to or by anyone or anything, for any purpose.

"Call in" means to make a phone call for any purpose other than a social one, usually to report or gather information. For this reason, we almost only "call in" to organizations. Like, if you're sick, you can call in and let your work know you won't be coming in. If a radio show accepts phone calls, you can call in to win a prize. If you've lost your credit card, you can call in and report it.

It's very rare to "call in" with friends or family. Nearly every call to someone you know has a social element to it. The only time you'd "call in" with family or friends is, for instance, if your father is in critical condition in hospital, and you call in with your mother every half hour to get an update.