Lead-acid batteries use sulfuric acid (H2SO4, a strong and nasty acid). They produce hydrogen gas when they are overcharged. Acid will only leak out of the battery if it is left on its side. If the battery gets knocked over, immediately put it upright, and wash your hands as soon as possible. Sulfuric acid is nasty stuff, and it's especially concentrated in a car battery. The caps on the battery will keep any splashing in (think of what is happening to the fluid in the battery when you're taking a fast corner in your car, or going over bumps: it's sloshing around).
The voltages inside the battery degenerate very very slowly as it is being discharged. It only shows changes of greater than 1.0 volts when it is approaching complete discharge. CAUTION: car batteries, though they only produce 12 volts, have absurdly high amperages. A short creates a pretty exciting spark, and if you were to be electrocuted you stand a good chance of dying on the spot. Do not let any metal whatsoever touch the battery terminals (unless of course you're connecting it, when you should expect a decent spark).
The UPS control unit might just think that the car battery and the stock battery are identical. They both supply the same voltage, it's just the car battery is bigger, so it has more sulfuric acid in it, which translates into a longer run time (more amp-hours). Accordingly, it would also take much longer to recharge. The higher amperage of the car battery would be appropriately restricted by the UPS control unit, I'm guessing. To further increase the run time of your system, you could wire the batteries in parallel (not series; series increases voltage and keeps the same amperage, parallel keeps the same voltage and increases amperage), but it would take forever and a day to recharge them.
I too would recommend using a more expensive marine deep-cycle battery for use in the backup, as this is basically the exact duty cycle this kind of battery is designed for. Car batteries are indeed designed to run the interior lights and the radio for maybe 30 minutes, then start the car, then recharge during your 15-minute or longer commute (lots of short trips, like under 5 minutes, are bad for your battery and your car in general, for many reasons; the battery doesn't fully recharge, and the moisture in the car's oil isn't fully evaporated off, which degrades the oil).