The thing that jumps out at me as sounding off is the use of nor. While nor isn't always used with neither, it usually is, and I think it would sound better with neither, or with or (since the "no" is a negative that carries through)
Moreover, since there is neither class nor office time...
Moreover, since there is no class or office time...
I also agree with user3169's suggestion to change "I extend" to "I am extending".
As far as your other specific questions:
Class is perfectly fine. But if you did want to say classes, you'd have to do some other rearranging ("since there are neither classes nor office times...").
You don't need "as well" at the end, but it makes the message clearer. Without it, it seems like one of the two dates given might be incorrect, because it would be a little strange to reference the same date twice without acknowledging somehow that it is the same.
So final full sentence:
Moreover, since there is neither class nor office time on Friday, I am extending the deadline for problem 1 to Sept 1st as well.