SOLUTION BELOW, but first, some discussion. 
You’re not the first one, there’s this:
Turning off notifications in game.
I can completely understand you, no excuses needed, and I must add that those notifications are truly irritating.
I don’t find this dramatic at all, your description is rather apt.
Indeed.
However, I’ve got good news for you: there is at least a partial solution to this problem. You’ve got to engage the ‘Busy’ mode (status) in the drop-down menu, just under your BattleTag, in the top right corner, where the (docked… or un-docked? Alright, it’s docked, apparently) friends panel is also located. I’ve noticed that sometimes it’s bugged and doesn’t work properly, that is, some notifications (‘toasts’… perhaps those are the infamous disguised ones ) about a friend going online are coming through (restarting the Battle.net app regularly seems to help, although it’s uncertain), however, it takes away most of the regular annoyances and the worst offenders. It mutes the in-game chat, too (including the obligatory annoying message sounds), but you can still type there (without seeing any replies or messages in-game) or use the app to chat while this solution is engaged.