Before going too far, you might take the batteries to an auto store and have them tested. That's an extended storage period. (And consider a lead-acid battery in-service should be replaced every 4-5 years.)
a) How can I charge the batteries using the solar panel?
There are probably tutorials that explain how to properly charge a lead-acid battery. The fact that the charging source is a solar panel probably doesn't matter too much (except when it is not operating at full efficiency.) Probably don't have to worry too much about the rate at which the panels can charge, just make sure you have a way to stop charging once the batteries are full.
b) Can I safely regulate the battery power (once charged) and power my arduino off it?
A 12V Lead Acid battery is no different than any other battery. Just remember that they can provide significant
current capability. In the event of a circuit/wiring fault, you'll be glad you are using some fuses.