Golok, if you think about it, making a sword as opposed to just about any other piece of smithing is a really time consuming and difficult project, and one that is most prone to error. You are attempting to make a long and narrow piece of metal, by the time of the Vikings it was largely carburized iron, known as steel, that was to be able to take any number of repeated stresses while being unsupported all the while. Further, it could not, contrary to your impression, be too heavy. Norse sword design was actually noted for lightnesss and quickness, the image in "The 13th Warrior" being pure Hollywood horsesh*t. Most of them run in the 2-3 pound (.9K-1.36K) range and have broad fullers down both sides of the blade to make them even lighter. Now, take this long, narrow, and light piece of unsupported metal and start banging it against things, as it was intended to be used, and see how long your average piece of smithwork will last before a flaw somewhere in it causes it to break, and you may take it as a given that any flaw will show up and will cause it to break. Being a true believer in Murphy's Law and in its First Corollary, i.e., "What can go wrong will go wrong" and "Usually at the worst possible moment", I see the flaw showing up in the middle of a battle. Can you imagine how bad off you would be to be in the middle of a fight, say a holmganga duel, and have your sword break. Friends who do battle re-enacting with swords tell me that it is a really bad feeling and they are not out there with their lives hanging in the balance!
So, as you can see, sword-smithing was not a task undertaken lightly, not if the smith wished to live and to prosper. It took a great deal of skill and a great deal of time and sweat to make a sword and all of this costs, whether in barter or in coinage. That is why swords were limited to those who could afford them, the wealthy, or who could earn them, the very brave and/or the very skilled fighters, at this time. One other way you could get a sword was to scavenge it off of the battlefield or to steal it from a grave, both of which are mentioned in the Norse sagas. In any case, the number of people with swords was always more limited than Hollywood would like us to believe throughout most of the Migration Era and of the Middle Ages for this reason.
Spears and axes, however, were easy to make and almost any village smith could do so. Further, they did not require much, if any, expensive steel the way that swords did. So these were the weapons that were seen most often. A spear could be any sort of point, needing no real steel in it, attached to a long pole and was as useful in hunting as it was in fighting and an axe was an iron head with a steel edge welded into the front of it and attached to the haft. It was something with which any farmer was familiar and was necessary for working around the farm, so they were as common then as a screwdriver is now. So those were what were used most often as weapons by the majority of the men. And they were used with a wide range of skill, just as any weapon is used with a wide range of skills today. Fighting with an axe is very different from chopping down a tree with it, as fighting with a spear is very different from hunting with it. Among other things, the tree and the deer don't usually fight back. A boar or a wolf does and that is why they tended to use special spears for them, but the general run of game was hunted with plain old spears.
As to sword skills, most who had them were probably skilled in their use or they would not have lived very long with them. A sword was something that attracted the attention of others who eyed it with some interest and desire, so it probably was a good idea to know how to use it if you carried it, just as it is a good idea to know how to use any weapon that you carry today. Carrying a sword was, at the time, an advertisement of your status as a warrior and as a man of status, given the sword's value and status.
I hope that I haven't gotten too obscure with this.
