Who cares what Dwarves used, they were such tanks it didn't MATTER!
I mean am I the only person who thinks the 500 dwarves armed to the teeth with the finest weapons available in the world would cut through the soppy Laketowners and the wimpy Wood-Elves like butter? 13 dwarves alone cut through to Bolg's bodyguard after all... Haha
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Anyway, to contribute to the Roman debate, Roman warfare was designed for Men and would not suit Dwarves. It was designed to simply be as efficient as possible on a grand scale. They didn't have the finest weapons or armour available but these items were mass produced and protected the most important parts of the body. Tactics were to minimise casualties and maximise kills but tactics were very generalised. However the Roman machine's best attribute (in my opinion) was the logistical system including road building, and scouting/camping tactics. This is what really gave the Romans the edge, their efficient administration overall.
However with Dwarves it's different. The dwarves have a small and slowly growing population. Dwarven warriors are specialist troops covered in fine armour and weapons instead of mass produced mediocrity (obviously if the battle is lost the enemy gets all the armour and weapons). They can't exactly afford to fight wars of attrition with a few battles lost. So the axe is designed to take out the enemy with one stroke and win battles relatively quickly.
And dwarves were really freakin' tough. So I think they could take some beating under chainmail that would usually damage organs. Not to mention deal with a scratch from a sword which would make advantage of the lack of Dwarven shield. So axe = awesome dwarf weapon.
As for whether or not dwarves could shoot bows, bit of a stupid question. French knights didn't just learn how to ride with lances and English longbowmen didn't just learn how to draw back a bow. A lot of the time they would have to adapt for the circumstances.
Swords - too short and can't pierce armour or cut through tendons that well. They were designed for slashing at necks and torsos (dwarves are too short) or stabbing (dwarves have not got long enough reach). Also swords are all metal, while an axe is half wood. An axe saves precious dwarven metals.