According to Oxford Advanced Learner's Dictionar of English, "service" is a countable noun when it means an examination of a vehicle or machine followed by any work that is necessary to keep it operating well: I had taken the car in for a service.

The above explanations confuse me.

Are the following sentences okay?
1. When our oven broke, we called a repairman for a service.
2. I had taken the car in for service.
3. That company provides after-sales service/after-sales services/an after-sales services.
4. The same as 3. in sentence structure but "good" is added before "after-sales."

The thing you describe as an examination followed optionally by the necessary repairs might better be described as a "check-up."  They'll figure out how to get your money anyway.

I guess it's probably true that what used to be a 60,000 mile check up is now a 60,000 mile service, especially if you take it to the dealer.  They'll have a bunch of expensive stuff to do whether you need it or not.

I haven't seen any lately, but they used to have computerized diagnostic centers which would check your car and tell you what's wrong with it.  Then you take it somewhere else to do the work.  It was a great way to check out a used car.

But the expression, "I took it for a service" is rare.  Little old ladies from Pasadena might.

If you're really flush with cash, you might say, "I took it for the 60,000 mile service."

1. When our oven broke, we called a repairman for a service. Nix the "a."  You might say, "for a service call. "

2. I had taken the car in for service.  Good.

3. That company provides after-sales service/after-sales services/an after-sales services.  You have an agreement problem here.  "An after sales service" might be a one-time courtesy visit.

4. The same as 3. in sentence structure but "good" is added before "after-sales."

"The company has good after-sales service."  That's fine.