Welcome to the asylum. Eagle eye? I happen to be an Eagle Scout, so I will give you "my" view.
1. Yes. There are four different colors which are dedicated to different functions. Brown is constant on for taillights. Yellow is left stop and turn. Green is right stop and turn. White is chassis negative return, commonly called "ground" although obviously it is not a true earth ground.
2. Yes. Of course, Your Mileage May Vary. If you are wanting higher draw things like television, microwave, coffee maker, air conditioning... you will need a LOT of battery and that will either get heavy or expensive.
3. Yes. Again, Your Mileage May Vary depending on how much stuff you want to power. If you only need to charge batteries and maybe have a few usb ports for a phone you can have a really simple shore power system that uses a standard 15A household plug and you would generally have a single circuit in your trailer. Many here have accomplished this with an extension cord and a power strip. The power will go to an AC to DC power converter to supply proper DC current to charge your trailer battery. If you have all those high draw electric toys you can choose to have a 30A system in your trailer. This is slightly more involved and has a special male plug that mates with one of the three female receptacles typically found on a campground shore power post. You could have as many as four circuits but the total of all the appliances you operate at one time can't exceed 30 amps.
4. Maybe. You can't really accomplish this within the 4 wire flat connector. You can do it within a 7-pin RV connector which also gives you the ability to have electric trailer brakes, but that isn't the best solution either. (If your trailer will be near or over 2000 pounds you might want trailer brakes anyway, and then you will need the 7-pin connector but that's a different conversation.) The better way to charge trailer batteries from the tow vehicle is with a separate connection using heavy cable (like the cable that connects to your vehicle starting battery) rather than the thin stuff you power trailer lights with. A connection like this is often made with an Anderson plug, but there are a handful of other great options.
Good luck and happy building!