What do you do with the tiles you pick up from the well and can't use? I know they count against you, but do they go back into the well or do they stay into your hand?
They stay in your hand. I'm not sure if there are any official rules available online, but the drawing-from-a-common-pile mechanism, shared by many games (e.g. Scrabble, but also Ticket to Ride) permanently puts tiles/cards in your hand. If it were differently for Triominos, the rules would explicitly mention this.