Luxon and Kurzick Armor

Is it possible to get Kurzick armor on one character and Luxon armor on another?

Yes.
You can also get both Kurzick armor and Luxon armor on the same character.

It is, but there's a catch; you must have more Luxon faction to get Luxon armor and visa-versa for Kurzick. There's a fairly easy way to resolve this, though; get 1000 faction, then trade it in for amber or jadeite. Now you lower your faction rating and if the other is more as a result, you can buy the armor.

Trading in faction takes faction away from the other side. If I have 1000 Kurzick and 600 Luxon, trading in that 1000 will also take away up to 1000 Luxon, 600 in this case.

Just get rid of it somehow and do a quick repeatable quest or something ^.^

So it has nothing to do with your guild? Also how do I increase my faction?

You could just get the blessing in the appropriate area and kill some stuff too once you've cleared out your faction. Though that does cost a small amount of gold.

Any quest that gives you the appropriate faction as a reward works too. AB for the correct side as well.

You only need to have the Faction you want to buy from 1 point higher than the other. So 0 in luxon and 1pt in kurzick will let you buy armor from a kurzick armor crafter.

Quests from the appropriate side will have a faction reward, along w/ any gold or xp. You can also get faction by going into an area that is luxon/kurzick controlled, and paying the priest of that side for a blessing, which will give you faction points for every kill while in that area.