2) Give us information that the RT game has started. Either play a sound (pretty much any "alarm"/ "timer" site can do this even with inactive tabs) or at least send a notification.
This is a thing if you play the mobile version. The game gives periodic, in-game notifications for how long before a RT game/tourney you entered is starting. From what I've seen, it gives warnings when you first join the lobby, then at 15 mins left, 5 mins left, and I think 1 min left. I do agree that this is something that should be in the PC versions as well.
Also, since JK mentioned Muli's userscript above, if you have that, that script does give a live countdown to when your next RT game/tourney is starting. From what I've seen, you have to be playing in full screen, doesn't show in split screen