Honestly, I believe switching characters after losing promotes degeneracy online. I've had so many matches where the opponent would lose the first game, then make me sit through all 90 seconds of rematch and character select screen just for them to hidden cursor counterpick. Even worse if your opponent wins the first game, you switch to a counter and win, then they switch to a counter to your counter. If it were up to me? Just pick a character before the set and just play.
If someone makes you wait for the character select to time out, take it as a compliment; they're acting petty and childish, which means you rattled them and you have the mental edge. Don't let them rattle you back; that's what they want. Just get in the habit of practicing some deep breathing while you wait, or distract yourself with something else.
As for counter-picking your counter-pick, that's just a known privilege that goes to the person who won the first match. Not meant to sound cold, but if you want to stop it from happening, you've got to be the one who wins first.
Agreed that the first pick should be blind though. I think it's fine for you to see your opponent's name (you should get the advantage of your player knowledge), but you should select your first character on different screens. It might also be interesting if you get to see each other's selections (now locked in) before picking your variation though; that incentivizes you to think about different MUs in advance and be prepared with different loadouts.