Well, it's been two days and 34 levels as a neophyte Mistweaver, and I've made the following observations.
1. A monk in full heirlooms levels INSANELY fast. Monks get a daily quest at the Kun-lai Summit monastary that grants a +50% bonus to experience for two hours. Though I haven't number crunched it, it appears to be cumulative with all other experience bonuses (heirlooms, Rested XP, Adventurer's Journals, etc.)
2. Mistweavers have a relatively small direct healing toolkit until the 30s. The majority of your healing is done through Eminence. Eminence is granted by the Mistweaver stance Stance of the Wise Serpent. If Eminence isn't enough, you also have Soothing Mists, a channeled heal that is usually more than enough to top up a wounded target in a few seconds.
3. Being able to heal through DPS is invaluable for soloing. The Mistweaver is amazingly self-sustaining when it comes to both health and mana. You can maintain a lot of questing momentum with virtually no downtime.
I've only had one instance provide any issues so far and that was Stormwind Stockades, and only in Randolph Moloch's room, if the tank pulls the whole room. Soothing Mist just wasn't enough to keep up a Bear or a Monk tank. The Bear threw a fit, but the Monk just agreed to pull more carefully, and we completed the Stocks very quickly after that.