Sewage coming out of concrete in basement floor
I think that it is very like to almost surely that your basement is related to the city sewer drain being full. Note that while it is often referred to as the sewer or sewer system, it does not contain sewage but rather it is for rain water runoff. The system that carries waste water from your home is (depending somewhat where you live in the world) a completely separate system.

So the good news that the water in your basement is probably not sewage but just rain water. Which explains the lack of smell. The particles that look suspicious might just be dirt or other debris from the basement (these can't go through concrete anyway) 

If you have not contacted your city about the water situation then I would do so, and tell them the situation and that it's flooding your basement.