I went 3-1 with AT yesterday and 4-0 with AU just now, so it's definitely possible to do well with non UN decks. And, coincidentally, I went "only" 2-2 with UN earlier today having two Seal of Deepwoods in the deck.
As for being level screwed, just remember that like in any other card drawing game, over time your opponents will get screwed as often as you.
And, yeah, it's almost always correct to play your level 2 cards during rank 2. This because you will then have a bigger chance of drawing level 3 cards during rank 3 and thus have a lesser chance of being screwed.
In general (without extra card draws or banishing), you're going to draw 2 out of 3 cards during a rank (i.e. draw 20 out of 30 cards over 4 turns). So, for example, if you have 3 broodqueens in your deck, you expect to draw 2 of them during rank 1. And, likewise, if you have 6 level 3 cards in your deck, you expect to draw 4 of them during rank 3.