People are generally happy with the Gaussian noise assumption. In some cases, there are acceptable linearizations of the dynamical model. One example is linearizing the quadrotor dynamics about the hover state (no roll or pitch). This approximation works well enough for basic flying, but you won't be able to pull off any flashy maneuvers because any hard bank will move you too far away from the linearization point for it to hold. A better choice would be a more complicated KF (I've used an error-state KF in the past).

The real draw of these filters, though, is that they are very fast. In my experience, most of the compute time every update cycle is spent on sensing because your sensors dump a ton of data that you need to process as part of your CV / SLAM / whatever pipeline (the outputs of these then go into your KF). The dream is to get a 10ms update loop so your control algorithms can do a good job, but this is easier said than done.

No, but nonlinear systems can be linearized about any particular operating point very easily, and the linearized system is often reasonably close to the whole nonlinear one.