Now I'd like to check if a customer with a particular customerId exists.
What do I do now? How do I name the endpoint?
REST is about state transfer. You transfer a representation of the resource between the client and server (REpresentational State Transfer).
You don't put domain specific logic into your URL scheme. A resource has a URL, and that is. You don't use URLs to start running procedures on the sever (eg hey server check if this customer exists)
That is moving into RPC territory, and is the opposite of what REST is trying to give you.
Why would i want ALL the details of the customer if all I want to know is if he exists or not?
REST doesn't concern itself with "why" the client wants the resource. That isn't for the URL scheme to manage. The client might want the resource just to see if it exists, it might want to see if the customer is from Washington, it might want the resource to see if the customer joined in the last year, etc etc
Your URL scheme doesn't care. This is on purpose, cause imagine if your URL scheme did care. You would end up with URL scheme expressing some sort of domain specific query language.
It is far easier to just get the resource and let the client figure out what conclusion it draws from the current state of the resource.
If you have a resource that is massive and your client would like the representation of this resource in another format, you can use the
Content-Type headers to negotiate between the client and the server a format that the client wants. For example you might have a representation of the Customer that has the entire customer data, and one that just has ID and name. The client can say it accepts this short format and the server can return it.
Though to be honest unless your resource is massive I wouldn't even bother with that.