The answers to these questions are going to vary a great deal between different periods within the Middle Ages. In particular, not just the form but the availability of arms and armour changes a great deal from the early medieval period to the later medieval period; this transforms the equipment of the late medieval soldier.

Since I am a student of the later middle ages, (the age of plate armour), I will focus on that period. The answers can and will differ for the earlier Middle Ages. Specifically, I am speaking about the period from 1350 to 1520.

So, to answer your question:

• After 1400 the pollaxe was one of the most common weapons used by men at arms (knights and those who fought like knights) fighting on foot, if not the most common weapon. With full plate armour to protect them, shields were not necessary, and with plate-armoured opponents to face the (still limited!) armour piercing abilities of the pollaxes's top spike (to find and attack the gaps of armour) and back spike (if the pollaxe had one this was one thing that -could- pierce plate armour, though only if it struck at the right angle). I right more about pollaxes here and here. In contrast, I see little evidence for the widespread use of one-handed axes; there are a few surviving, so they were not unknown, but compared with two-handed pollaxes they are not common.

• In the later middle ages most spears used on foot were two-handed, and polearms like pollaxes and halberds and partisans were exclusively two handed weapons - it is impossible to make a slash with a halberd with one hand and control your weapon! The late medieval spears that I am a best acquainted with, the 'long lances' of 100 years war men at arms and the pikes of the Swiss and others, were two handed. That being said the ordianances of Charles the Bold also require that pikemen carry a target (a medium-sized shield, often round), so being two handed didn't preclude all shield use. I am less acquainted with the spears used by Italian infantry in this period; in art they are sometimes shown wielding spears one-handed and carrying shields, though in some case this may be a classical affectation by the artist. On the other hand, cavalry spears were used one-handed, whether they were heavy 'knightly' lances, medium-sized cavalry spears or short javelins.

• The usefulness of polearms in individual combat depends upon the polearm! Pikes are useless and individual or small-group weapons, while pollaxes (being shorter and more maneuverable) are excellent weapons for fully armoured men, whether they are fighting together or alone. Halberds are somewhat in between, since their longer shafts make them less maneuverable than pollaxes. However, keep in mind that late medieval fighting (and fghting in other periods!) placed a premium on staying together and 'holding the line' - even in cases where formations weren't as drilled and regulated as they were with the Swiss later in the period. Generally speaking truly individual combat was for the tournament and judicial combats on one hand, and for civilian fights on the other (being set upon while travelling, for instance). In civilian life, a sword was much more useful (and handier!) than a pollaxe.

• Swords were no reserved for the nobility. We shouldn't think of medieval society (least of all late medieval society) as being so stratified that certain arms and armour were socially prescribed. Now, in the early middle ages swords were quite expensive and relatively rare, so they may have been more likely used by the wealthiest (which in the earlier middle ages would be the aristocracy). But swords become more and more common as the Middle Ages progress, until in the later middle ages they are quite common indeed - by the later 13th century there are schools of swordsmanship that are taken up by common people, and there are frequent complaints that apprentices, students and other young (and not very rich) rowdies are causing havoc with their swords. I write about Swords and Society from around 1250 to 1600 here.

• As with swords, we shouldn't think of armour as being socially prescribed only to people with titles. Now, full plate armour was expensive and the people that wore it were drawn from the wealthier parts of society. But this is a much broader class than 'the nobility', which is a narrow elite-of-the-elite. In the later Middle Ages the vast majority of people wearing full armour did not even have the title of 'knight' much less 'marquis' or 'viscount' or 'baron' or other titles of the upper nobility. Instead they were from the wealtier portions of 'the commons' - though they often took on the quasi-aristocratic style of 'gentleman' (which is not the same thing as 'noble') and are often referred to as 'esquires.' But even this undersells the social mobility that could happen, at least in England - in the hundred years war we see people recorded as archers in one personell record becoming men at arms in another. We see elder brothers as men at arms and younger brothers as archers. The grandsons of peasants might become knights, as happened with the Paston family. While society was definitely organized as a hierarchy, the lines between it were much more fluid than we might assume, especially when people could pursue avenues of social mobility like the Church, the Law and soldiering. Now, with all that said, there were a lot of common soldiers in medieval armies that did not fight as men-at-arms in full plate armour. In much of Europe they would be the majority of armies. They would wear partial armour, or no armour at all. I write about their equipment in these posts:

• English Arms and Armour in the 100 Year's War

• English Arms and Armour during the Wars of the Roses

• Armour's affordability in the 15th century, and its use by infantry