I haven't soloed any of the HW dungeons and won't even think about it until I am at least level 70. Even then, I would want to be at a very high ilv. I have soloed every ARR dungeon that can be, as some cannot. Basically, if the boss has a mechanic that requires more than one player in order to clear the fight, then it cannot be soloed if your DPS isn't high enough to kill the boss before reaching that phase.

I've soloed dungeons as SCH, WHM, and DRK. I really can't say which is better. I simply adjust to the abilities of the job I am using. Surviving a wall to wall pull is different with a healer than it is a tank, but I wouldn't say one is better/easier than the other. However, bosses are more fun to do as DRK.