Airplane mode? Voicemail? Where does it go?

Sure, they are stored on whatever server email service you are using. It's the same thing as turning off a desktop and then rebooting later. Once rebooted, the mail streams in.

Voicemails and text messages will appear automatically when your phone connects back onto the network. This is how any mobile phone works. It's the same as is your phone was off and you turned it back on. They are stored on the network until your phone is able to retrieve them. You will only lose them if your phone does not connect to the network within the expiry period for the messages, which is usually many days.

Since you check your email manually you will have to manually check your email once you are back on the network.

when you are in an airplane mode, it is stored in a server, and once you release the airplane mode, the reception will tell the server that you are back on and you will then receive text and voicemail.