1 Tank, 1 Support, 3 DPS. The individual classes don’t make much of a difference. Given the same skill level, Mages will hit the highest DPS, followed by Rangers/Scoundrels, then Shaman. You can absolutely do all content with any mix of the classes though, so long as you fill the roles.
Warriors tend to have better single target aggro management, whereas Paladins have better group aggro retention.
Bards don’t heal as much as Musketeers, but provide buffs for your DPS that make fights go significantly faster.
I believe @NotScott did a whole thesis on what rotations hit the highest DPS though. I’ll just leave that tag there and let him do his thing 
Some of the best Shard Dungeon runs I’ve done have been when we were all goofing off in chat. Callouts like “hotfoot reminder”, “[MECHANIC HERE]”, etc can help, but it depends on the group. I’ve done Shard 15s where nobody talked, and I’ve done some where we were discussing some raunchy subjects. It just comes down to the players you’re playing with
Unless you’re a Tank, focus on your weapons more than anything. Armor on Cloth/Leather classes don’t protect for much, especially in high level content. Each +1 on your weapon increases damage by around 5%, so get that +6 as soon as you can.