In, re. Starting an A/C with an inverter.
Most air conditioners need about five times their running load to start. So, if your air conditioner draws 500w when the compressor is running, you will probably need an inverter of at least 2500w to start it.
Will a 2000w genset start it? Depends. Most conventional generators can take a surge by simply allowing the prime mover to slow down for a few revs. Will an inverter/generator like the Honda or Yamaha do this? I have no idea.
An inverter cannot sag like this, so it must be sized to handle the entire load. To reinforce dwh's point, a Magnum inverter/charger, with the hybrid power feature may be able to protect an undersized generator, but the inverter/charger MUST be sized large enough to cover the starting surge.
In my case, a 2800w Magnum would start a 6k BTU air conditioner. Never tried it with a generator.