Do you have all multiplayer DLCs installed? They're free anyway, so don't hesitate. And you'll get faster into games that way too since most of the userbase have them and you wont be able to join any lobbies with DLC maps, guns and characters... which is pretty much 95% of the lobbies i bet.
Or maybe try to reboot Origin? I dunno, maybe it has something to do with this. Unlocked characters shouldn't disappear like that. Most of the time when ME3 is acting strange(i used that have no sound sometimes...), i just reboot Origin.
If you don't see the Galaxy map with Galactic Readiness % in the main menu before going into multiplayer, or if the Galaxy map is all red when you are in the multiplayer menu, it's already a sign that the connection to their servers isn't working properly, and so your unlocked characters may not be showing. So try to reboot Origin if you see this, it could help. I played a lot these past days without connecting issues(well other than the usual "buying packs" one), so i doubt their servers are having problems.