Do airplanes with fixed gear usually turn upside down during water landing?
Depends on the airplane.  I recall reading about a guy flying across from the Aleutians to Japan, in a Piper Super Cub of all things, and he had to ditch it for whatever reason, which he was able to do next to a freighter.

To his surprise, it didn't flip over but kind of submarined, stopped, then popped to the surface nose down but upright. It's possible being a taildragger helped.  I think tricycle gear airplanes would be more prone to flipping right over because the pitching moment from water contact is a little stronger with 3 gears hanging down vs two.

I expect though, that if you surveyed ditchings you would find some do and some don't.  Pitching moment from water contact, elevator effectiveness in resisting the pitching,  amount of up elevator at water contact, speed at water contact, maybe the surface contours of the fuselage could promote it or inhibit it, all kinds of variables.  

As xxavier says, it would be prudent to assume you are going to flip over, and if you don't, bonus.