If you’re both on XBox, I’d recommend using the XBox party chat over in-game chat. In some games (including BL1 from memory) the in-game chat is silenced during loading screens and map transitions. (Not sure about BL3 in that respect - I’ve only played coop with the in-game cross-play feature once so far).
For the odd audio, setup at each end plays a big role. I’ve noticed similar effects from time to time through XBox party chat when the person on the other hand is close to their speakers and/or has the game audio turned up loud. (You’ll also hear yourself and their game audio on a slight delay if they have the chat going through their speakers rather than through a headset.) That’s actually a physical system/setup limitation, and there’s no easy way for chat systems to get around that.