I've noticed a fusion of two weapon types will always produce the same weapon type, so I assume there is some sort of system for determining the weapon type produced, as opposed to it being random.
What determines what type of weapon will be produced through a weapon fusion?
You are right, this is a predictable system. In order to figure out what weapon you will get, you will need to know a few different things in order to figure it out.
First, you need to know what weapon combinations will result in the weapon type that you are going after. This (translated) chart will help with that:
From there, you need to know what “number” the weapon you want to make is. There are 12 of each weapon type, and they are numbered from 1 to 12. This chart lists the weapon numbers for each of the weapons in the game: 
When you fuse two weapons together, you just add those numbers together to get the number of the resulting weapon. If the result is higher than 12, it just wraps back around to 1; in other words, if the two weapon numbers add up to a number that is higher than 12, subtract 12. (For example, if you want to make a Palutena Blade, you could use the Cragalanche Cannon plus the Dark Pit Staff - 12 + 11 = 23, 23 - 12 = 11.)
If the fusion results in just an X and a ?, it means that you haven’t unlocked the weapon in-game yet, and you can’t just fuse things to create it. You will need to explore the game more completely, finish more levels, etc. to unlock all of the available weapons, but then you can fuse things to your heart’s content.
This is not entirely foolproof - sometimes you might randomly get something you didn’t expect. it does work most of the time, however. Also, there hasn’t been a way (as of yet) to know what stars and abilities get carried over when you fuse weapons, unfortunately.
