They store all the settings (easy, hard, balls/play) as well as the audits (# of games played, # of extra balls, # of specials) plus of course scores. On a system 11, with dead batteries, it should flash "Adjust Failure" if the batteries are dead and force you to go into test mode to get past it.
I own 3 Sys11's and currently have a friend's Taxi here as I fix a few things for him. Given what you've described, and the clueless seller (or outright deceptive seller) I'd say $200 sight unseen (to me) is more than fair, $300-$400 tops if you are okay with playfield condition, plastics, plasmas and mechanics underneath.
Some sellers see listing prices on ebay = value vs. the actual sold auctions indicating value. IPDB is wrong, more than 200 Marilyn's exist, and is never gonna be worth $7K as they claim. Use your knowledge from actual closed auctions of working vs. non working from eBay and PinPedia.com's results to learn value.
On a Sys11, the CPU board does the brains, some sound, all solenoids/flashers, switch matrix and lamp matrix all in one. Untested they go for over $100, fully tested working $300 typically. Then there is a power board for GI, CPU and plasmas, then another board for solenoid power, then the plasma driver board which with 2 working plasmas run $125-$175 typically, plus a sound board that runs about $50 typically. So there alone you're at $500-$600 if the 3 main ones are missing or bad, added to $800 makes that a bad investment before you even start looking at plastics and playfield.