The most likely reason you are getting texts while in airplane mode is that they are coming through a Wi-Fi connection. It’s also possible that the phone isn’t actually in airplane mode or that airplane mode isn’t working correctly. When a phone leaves airplane mode, all stored messages will come through quickly.
If you turn airplane mode on, the other person won’t know you saw their snap. As soon as you turn off your airplane mode, the other person will be notified that the snap has been opened. However, you can trick Snapchat with airplane mode into not letting the other person know that you have opened their snap.
Callers will hear your phone ringing and think you’re available. Although you will not hear the phone ring, the caller may receive an unavailable message. If you have voicemail on, the caller can leave you a message. If the caller does not leave a voicemail, you will never know that the person was trying to reach you.