Congratulations on the new to you Swift and welcome to the JOF.
You have a whole bunch of questions and a mess of stuff to check on your TV (tow vehicle) and TT (travel trailer).
You should find a sticker in your TT giving figures on its gross weight rating etc. Your TV will also have a payload figure showing the maximum load it can carry (= in the cab + in the bed + the hitch + TT tongue weight). TT tongue weight should be in the range of 12% to 15% of your TT gross weight.
Using your VIN# and google, you should be able to find your rear axle ratio and more on line.
One thing you need to check carefully is the roof and inside all the cabinets etc. for any sign of water leaks.
If you are plugged into a standard home outlet, you can't run the A/C. But it should charge your battery, run your refrigerator, and (I think) your microwave. You should have a panel of circuit breakers for your TT AC power and fuses for your DC power. From your photos, it looks like it is under your refrigerator. That panel controls the power to your AC devices including the TT converter (converts AC to 12volt DC) and also charges your battery.
You also need to be sure you have propane for the stove, oven, furnace, water heater, and the refrigerator - when shorepower AC is disconnected.
Yes, here in AZ you should be able to connect to the 'city' water connection any time, as long as freezing weather is not expected. (I'm further north so won't be connecting water to mine for a while longer.) It is recommenced to use a pressure regulator to keep high pressure water from damaging the TV. If you don't have one, you'll need it.
Did the owner/dealer mention anything about the water heater bypass? Before you turn on the water heater (on propane or electric) you'll want to be sure the heater tank is full of water. With water at the 'city' water connection, the 'cold side' faucets, showers (inside and outside), and the toilet should all have water. Due to the water heater 'bypass' valves, you may not have water out of the 'hot side' at first.
If you fill your water lines, then you need to watch for freezing temps if you are going camping at higher elevations this spring.
If this is your first TT, you'll need wheel chocks and leveling blocks of some kind.
Check all tire pressures and inflate each to the maximum shown on the tire's sidewall. And, by the way, check the age of your TT tires.
That's a start....
Sherm & Terry w/rescues Steve (std Poodle) & Eydie (min Schnauzer)
2015 Jay Flight 27RLS (Camped: 102 nights 2015, 90 nights 2016)
2006 Ford F350 6.0 PSD, Lariat, 4WD, CC, LB, SRW, auto., Camper pkg.