will only doing kills/group quests get my team to level 80 with only rested exp and no heirlooms?
From my experience on my first team (5 shammys) I would say no. I ran out of kill quests around 74 and that was even from playing abit of BG and rested xp. Eventually you will have to do collection quests and imo you should just bite the bullet and do most of them, especially ones that chain to other quests.