Do per connection random keys solve the stream cipher key repetition issue?
I'll answer per related question:


It seems like you are trying to describe a scheme that relies on a One Time Pad (OTP). To use an OTP it is required that your scheme complies to the security requirements of a one time pad. If it does, it will be provably secure (except for the leaking of time and length related data).
Now you are trying to create a slow, untested stream cipher out of your OTP scheme, because you do not comply to the OTP requirements. You are better off choosing a well known, fast stream cipher instead.


So choose a fast stream cipher and try and create a good key management scheme around it; you haven't described how you distribute your session keys.