Trying to figure out what condition a car is in is like being in an episode of Law and Order. "this doesn't look like the stock battery". You could get a million useful responses back from that. "yeah, this thing eats batteries once a year", or "just put it in, so it should be good for 5 years", or "I was jumpstarting this guy when the old one just exploded!". The thing was probably just old, but it doesn't hurt to ask.

About the company owned thing... 10k a year isn't bad. I just don't like the idea of someone driving (and maintaining) a car that's not theirs. How did you treat the last car you drove that wasn't yours? I know that my last Magnum rental car didn't like neutral drops....