As a quick check on an invertor, look at the label for the current or amp draw on the appliance, multiply that by the voltage and that will give you the nominal wattage then add 10% for surge. This works in most cases except for electric motors which have an instantaneous start draw of 6 to 7 time normal amp load then you add 20 to 30 percent.
As far ar the lap top you can think about it this way, The transformer drops the voltage from 120 to about 9 or 12 depending on the manufacturer the sine wave of frequency response unless there is a great difference will make no difference, the true voltage is the killer.
Well, anything which works off of DC, has the input voltage clamped and filtered, so 'messy' waveforms should not be a problem. However, things which use the power directly could have problems. Didn't seem to hurt it, but using my tube TV/VCR gave very poor performance. Also, Flourescent lights flickered and buzzed.