Hey Breakthroughs,
Thanks for using Apple Support Communities! We understand you're experiencing an issue with missing messages, and listening to music when not connected with Wi-Fi. We can definitely help with both of those!
With your messages, are you only missing your text (green messages) or are you unable to receive iMessages as well? If you're not receiving iMessages check out what to do here: If you can't send or receive messages on your iPhone, iPad, or iPod touch
If this is isolated to text messages only make sure in Settings > Messages that you have send as SMS and MMS Messaging enabled there.
Another setting to check out for Messages is in Settings > Messages > Keep Messages, and ensure that your preference is set correctly there.
With your music, check in Settings > Cellular and ensure that Music is enabled under Cellular Data. See more about that here: View or change cellular settings on iPhone
Thanks again, and have a great day!