How do I maintain lithium batteries for maximum life? Keep them charged up all of the time or let them go down? I don't use them often at all. Does it hurt to use 12 volt tools with 18 volt batteries occasionally?

Do not let Lithium batteries shelf-discharge to a low level. Lithium technology happens to be good at holding charge a long time sitting on the shelf, but they do need to be charged occasionally. If I haven't been doing projects that prompt me to charge them, I cycle each battery on the charger every 6-8 weeks.

I would not use an 18V battery on a 12V tool. You're asking to fry the tool if the regulator can't handle it.