Welcome. I would check the FAQ's section and wiki as there is a ton of information for newbies like yourself that will answer many of your questions.
I as well as many will strongly suggest you avoid DC/AC inverters. The reasons can be seen in the FAQ's section but basically they are very inefficient and usually don't survive crank. Not to mention all of the stuff you are trying to run is DC to begin with so you are converting from DC to AC to DC.
no, they don't have batteries (I won't say all but certainly most)
You can split it as many times as you like so long as you have enough wattage
Use a relay. Take a high power fused line off of your battery to a standard auto relay and trigger the relay with the ignition wire at your ignition harness.