Congratulations, and welcome to the Forums! There's an incredible amount of info here... poke around for a while and you'll undoubtedly be overwhelmed with the amount of stuff folks here know!
To answer your questions, there will be a set of gauges that will indicate your tank levels. They're not all that accurate, but they'll give you an idea (most of the time) of how full they are. I generally use the non-formaldehyde enzymes in my black tank, but there is no consensus. If you dump your tank and rinse it regularly, you may not even need chemicals. Then again...
The cord is a 30 amp RV cord. You can use the 110 adapter to plug it into a standard 15 or 20 amp circuit. Do NOT run the A/C on using that adapter. There won't be enough amperage to start your A/C and you could potenitally damage the compressor.
Turn the propane on and let it sit for an hour or so, and then try lighting the stove burners. Sometimes it takes a while for the propane to purge the air from the lines. All of the hot water comes from the water heater; usually a six gallon propane, or propane/electric setup.
There's no reason NOT to use the propane systems, as long as you don't smell propane anywhere after you turn the tanks on. If you do, shut the tanks off, and take it to a service center who can track down where it's coming from. The water heater, stove, furnace(s), and refrigerator will all run on propane, so it's a pretty vital system.
Ask questions of the previous owner who is delivering it... much of what you've asked about is easy to do, but difficult to explain in writing.
Read the many threads here on weight distributing hitches and sway control. Most good RV stores will be able to help you out with a proper hitch setup.
Good luck, and again, congratulations!
AIR 2053 Current: 2006 Born Free 32 RQ Kodiak Chassis, & 1995 Coachmen B-van
Former Airstreams: 1953 Flying Cloud, 1957 Overlander, 1961 Bambi, 1970 Safari Special, 1978 Argosy Minuet, 1985 325 Moho, 1994 Limited 34' Two-door, 1994 B190 "B-Van"