That's the simplest inverter circuit that I have ever seen! ;D
1) I can't picture what its waveform would look like. A spikey square-wave with its horizontal parts severely tilted?
2) Its frequency depends a lot on the inductance and saturation level of the transformer, a little on how much load and a little on the capacitor's value.
3) Ocassionally it might not start oscillating when powered, it would have both transistors conducting at the same time and blow its fuse or just sit there and smoke.
4) If it is overloaded or the battery voltage drops with a normal load, it would stop oscillating and blow the fuse or just sit there and, you know. ;D
5) Without protection devices, a transistor would probably blow-up if the load is disconnected or turned-off with the inverter running.
6) You could use NPN power transistors if you reverse the battery connections. ;D ;D