I've Googled for this but it's hard to know what websites can be
trusted. In here, if someone makes wrong statements there's an army
waiting to pounce!

I'd like an inverter to run off my car battery. I can't seem to find
its ampere-hour rating but it's 350 mm long and is the biggest battery
in any shop around here. There's also a V8 engine to charge it if need
be.
I want to run a bunch of 230 volt tools and appliances off the
inverter - one at a time.

An angle grinder (what else?!) rated at 500 watts.
Maybe a circular saw at 700 watts.
Maybe a water blaster at 700 watts.
When one of those is not being used I'd want to run a fridge/freezer
(probably under 100 watts), maybe together with a desktop computer
with CRT monitor and a Philips LCD TV.

There's a cheap inverter that says it's rated at 1500 watts continuous
(3000watt max). It will be something like a square wave output. So
really, will the above appliances run on that without damage?