All are grammatically correct.
1-3 are close in meaning but not precisely the same. In #3, "since" will be interpreted to mean "because".
In #4, we have seen him one or more times since he left.
#2 can be a useful sentence:
"I have been busy. I still have time to study English and take care of the children. Unfortunately, I have not been seeing you since war broke out."
In this sentence, "since" can mean "since the time" or "because". The precise meaning can only be determined with additional context.
#3:
This is a very good sentence since "since" can have the same meaning as "because". Thus, I would interpret #3 as
"I have not been seeing you because war has broken out."
It is more difficult, but not impossible, to make sense of #3 if you interpret "since" to mean "since the time". It would not be an error, just a bit awkward. If you want the meaning "since the time", #2 is better.