Honestly, the multiplayer is pretty much dead, plus it's not fun to play online, lose, then wait 3 rounds to start over or TRY and find a lobby, let alone one without some form of cheaters/ hackers.
Although, funny enough, if you win or complete a level in single player? It WILL count to the achievements! Your best bet is to just put the rounds on 5 or however many you want and play. I'm not sure if changing the times affect the achievement progress (probably not) , but yeah it's that simple! Got it done in about 2 or so hours (some of the levels are hard or a bit bullshit ngl).
Sometimes steam may not update you completing a level or something, so just play another round or 2 and it should kick in if it doesn't give it to you right away.