How many... it all depends on variety and growing conditions. I would just let the vine grow as many as it wants. My own exception to that is to remove any squashes that set after the middle of august, since they stand little chance of ripening completely in my climate. And I figure the vine's energy can go into ripening the other fruits. I can't say if it really helps or not.
Do not trim the vine shorter,. The more vine, the merrier. Hand pollinate only if your fruits are not setting. Or if you really feel like doing it. I sometimes do it just because.
It still seems to me a bit early in the season to be worried if fruits are setting, since often the first ones don't do very well.

I planted a disease resistant type last year, it was a smaller variety. I am unsure of the name (but bought seeds from Johnny's). I had two fruits per vine in the end. Bee pollinated. I only ended up with 8 in the fall. This year I have 8 vines, so I am only expecting 16 fruit. Mine aren't even flowering yet.