I imagine it had some more practical applications aside from combat but from what I know the primary weapon of most legionaires from early ones to later all used a form of short sword, most images of auxilia also have them use either swords or spears.
Was there a cultural or perhaps practical reason why axes were fairly uncommon?
Reasons such as most Infantry Rome fought were unarmored/lightly armored making a sword more practical?
It wasn't - at least, not during the Republic or the Principate. There are a few reasons for it, but it's hard to say which one was the greatest factor.
The Romans fought in a strict formation. Each man only had a small area to fight in, allowing their comrades to support them, and making those famed tower shields (scutum) that much more effective. With that small area, their weapons were incredibly well-suited to the situation. Their javelins (the pilum) would be used to disorganize and demoralize their enemies, or even as a conventional spear to fight off enemy cavalry. When the legions would close, however, their primary weapon was the sword (gladius). The sword, adopted from the Spanish, was the mainstay of the Roman legions, and it was basically perfectly well-rounded for the fighting they did. Close-quarters fighting? Well, the sword was best at stabbing (although, contrary to some modern belief, it was really good at slashing bits off of people, too!). Stabbing is a short motion, allowing the soldier to stay mostly defended by his shield, remain controlled, and not endanger his companions who are all close by. Needless to say, this advantage was not shared by the war axe.
The auxilia used similar equipment to the Romans they fought beside. Not only were most of the auxilia Italians for hundreds of years, but even as they became more diversified, they still used Roman equipment and discipline to their own advantage.
Culturally, the Romans were an incredibly traditional people. They had the gladius and it worked well for them for the better part of about 7-800 years. Why would they swap out for a weapon that was untested, less useful, and, in the Roman's eyes, less Roman? The Romans showed an incredible willingness to adapt to use whatever was necessary to face the threats at hand; the gladius itself was an example of that, as were the various evolutions of the Roman army, the very armour that the Romans themselves used, their navy was - if we are to believe our sources - entirely copied from the Carthaginian navy, etc. If something was more useful, they would absorb it rather quickly, despite their traditional attitude. The war axe was not an innovation on that level.
I hope that covered all of your points :) For more info, check out Adrian Goldsworthy's work - he's a brilliant authour, and his books basically outline the Roman Army as completely as anyone could ask for. More specifically, check out his The Complete Roman Army and Roman Warfare :)