Is there a way to get a notification that I'm off the air?
I don't think this is possible, but there might be an app out there for it. I'm not sure. I'm fairly certain that it's not a native function in Android though.
Is there a way to reconnect without rebooting the phone?
Airplane mode on/off seems to be the easiest route. If that doesn't work, you can try going into "More..." on the settings menu and look for available networks manually. Maybe you'll find that you find other networks, just not your carrier, and in that case it's actually a coverage problem, or a problem with their equipment since you can actually see networks, just not the one you want to connect to.
Is this something that happens to others, or am I just lucky?
Connection drops to the network is very common, but it's usually different kinds of drops for different reasons. If you have rooted your phone and you're running a custom ROM, you might need to check out GetRIL. If you're running stock, try a factory reset/reinstall of your device first, or see if there's any updates available from Samsung.
I get drops at times on my Galaxy Nexus as well, but I only lose data connection, I still see my carriers name and I can recieve SMS/phone calls. This sounds completely different from your issue if you are actually losing contact with the GSM network completely.
I'd recommend that you check out if you can find other carriers networks or not (mentioned above), if you can find others you need to notify your carrier first of all. If you can't, it's probably phone-specific but even there your carrier/Samsung should be able to help you if you still have warranty and the device is not rooted.