Often the gearbox-driven pump assembly contains both an LP pump along with the HP pump. The purpose of the LP pump is essentially the same as the tank boost/transfer pumps, i.e. to keep a positive pressure head on the HP inlet.
Any positive-displacement pump (whether gear or vane type) can cavitate under the right conditions. In fact a marine propeller can cavitate as well if driven hard enough.
And yes, gravity feed will work just fine if the flow demand doesn't exceed the plumbing capacity/vertical head/viscosity limitations. I know of one application that is always started by gravity feed, but switches to a positive LP boost before reaching idle. (There are probably many such aircraft...)