I hate to be the bearer of bad tidings but a 400 watt inverter is not going to be able to run much more than your trouble light
The 400 watts is probably the peak rating for that unit so it will run continous at around 350 watts FWIW a average skill saw needs around 1000 watts at no load. We use 5000 watt inverters on our maintenance trucks. You could probably get by with a 1200 watt unit, barely. You would want to make sure you run the truck once an hour or so to keep from totaly draining the battery. A small generator would be a better choice if you can beg, borrow or buy one. As far as running cords, the larger the gauge (smaller the number) the better to limit line loss. Personally I don't bother with cords under 12 gauge.