The MOV, as 1995brave says, is the gizmo that provides surge protection. They have a limited lifespan. They conduct when the voltage exceeds some threshold. (The ones they use in surge protectors actually trigger at rather more than 130V because the peak voltage of a 120V RMS sine wave is around 170V.)
It would seem that either the voltage output of the generator is high or it has higher-voltage spikes/noise.
Inverter size is determined by the power requirements of the stuff you're powering. Game consoles in particular can vary a good bit in power usage; Xbox's and PS4's are comparatively power hungry. At any rate, 400-600W should be plenty for a game console and a TV. You may be able to use a lighter-powered one, but it would be preferable to have it wired directly to the battery. Just the TV, or the TV and a DVD player, might well be OK with a lighter plug unit; modern TVs don't use too much power. It depends partly on the quality and length of the 12V wiring to the lighter plug.