They are technically rechargeable. Not very often and their capacity deteriorates quickly, but there are specialized alkaline chargers that allow you to recharge them a couple of times.
You can tell whether they’re full or empty if you drop them on a hard surface. Empty ones will bounce slightly and make a “brrrt” sound, while full ones will stop immediately and make more of a “tock” sound.
Some fun facts about alkaline batteries:
Don’t think it’s a good idea to apply blunt force to a battery, but neat
You’re only supposed to drop it from an inch or so. Just enough so you can tell whether it bounces or not.