Everyone has their own specific things they look for when deciding who they give commendations to, but I think most people can agree that those who ask for commendations generally won't receive them. People see commendations as something you earn based on your performance and attitude, and by asking for them it makes it seem like the person you were throughout the dungeon was fake.

As for who gets most commendations between tank and healer. Based off my personal experience, it feels like healers get more. I usually run dungeons with my girlfriend and for ARR and HW we were a tank/healer couple, and she has absolutely sh*t all over my commendation total almost 4x over (she is at 11k and I'm at almost 3k). It felt like we'd come out of an expert roulette and she'd have 3 commendations because even the SMN's pet commended her.