Quantum field theory and General Relativity don't appear to mix because a quantized spin-2 (read: gravitational) field can't be renormalizable (read: finite) in four dimensions.
There is experimental evidence of the quantized nature of the gravitational field.
The first would require a treatise. I'll see if I can search up a cogent link tomorrow.
As for the second, here's an old FR thread about an underappreciated demonstration of the quantum nature of gravity.
Specifically is a field or a wave quantized?
I don't really understand the question "is a wave quantized"; I might recommend that you read up on "particle/wave duality". As for fields, we know that the electromagnetic field, the weak nuclear field, and the strong nuclear field are all quantized, meaning that they can be modelled as the exchange of "force particles". We don't know mathematically how to do that with gravity, yet.