Depends on how big the solar panel is. If it's a small one, yeah you can use it to trickle charge. If it's got any wattage to it at all, you need a charge controller to avoid destroying the battery.

Your solar panel probably has a zener diode in it already - most of them do to prevent a battery attached to it from running the solar panel in reverse, but if it doesn't you'll need to put one of those in the circuit.

What do you plan on running off the inverter? Most of those cheap models don't put out s clean 60Hz 110rms sine wave, like most A/C powered devices like. If it's a toaster, it's not going to care. If it's delicate electronics, you could burn it out.