The answer to these questions will depend on several factors.
- Early or Late Era - Difficulty Settings - Ability on the battlefield.
Having only played one 6.2 campaign (and an early campaign at that), I don't have any direct experience but am going on what I've seen others talk about.
The easiest factions would generally be those that are in the corner of a map. England, Moors, Fatimids, though I would say in this case, the Fatimids might be an easy faction in early era. They have plenty of high income cities around them (Cairo, Alexandria, Jerusalem, Antioch, Damascus), and only one minor Christian faction (Templars) to deal with. The downside is if one of the catholics decide to call a crusade, you'll have most catholics after you if you become a target
The Kwarezmian Empire would be another strong early faction to play. You have tons of rebels to go after, and so won't need to piss off any other factions for territory, and in about 40-50 turns you can have about 20 regions.
As to which would be the most fun, that all depends on what you consider to be fun. If fun is playing an underdog against more powerful neighbors, try Aragon or the Templars.
Although I know "most difficult" wasn't part of the question, for a challenge you can try either the Khwarezmians on VH/VH or the HRE on VH/VH in the late era. In the case of the Khwarez, you've have 4-6 full mongol stacks coming right for you in the first 10 turns or so, and in the case of the HRE, you'll have all your neighbors looking for blood... Venice, Genoa, France, Denmark, Poland, all within the first 10-15 turns ;)
The key to the HRE VH/VH is to blitz each faction and take several territories before they decide to get aggressive on you.
Ultimately the best advice is to hang around the SS TWC forums, and you should be able to make some informed opinions.