What happens if you place Fire Damage and Frost Damage on the same weapon? Do they cancel each other out?

Really?

No.

nope.
The weapon will simply apply both forms of damage on the enemy.

A common use of weapons like this are enchantments with both lightning damage, and frost damage. You are dealing a massive amount of damage to their health while also dealing damage to their stamina and magica.

Here's an example of a high level sorcerer build that uses a weapon with both lightning and fire, also very effective. https://www.youtube.com/watch?v=LG-KgAjAq54

Hope this helped

Really? Shock and Frost/Fire are the better combinations? I wouldn't of thought that they would have such an extreme effect!

I thought this was going to be about something like the Ring of Desiccation from Oblivion. Water Breathing + Water Walking.

Is it even possible to make a conflicting enchantment in Skyrim?

I guess it depends on your play-style. People believe frost to be a sub-par form of destruction magic because a majority of enemies have high frost resistance. I prefer it only on weapons I will multi-enchant for this vary reason.

I prefer it on these weapons that will have multiple enchantments, however, because they will regardless deal massive damage regardless if they are quite resistance to frost (base weapon + fire/lightning damage + frost damage: Thats basically three hits in one).

Now like i said, it depends on your play-style. I personally enjoy forcing my enemies to use nothing but raw attacks (depleting them of both magica and stamina). This is why I prefer ice + lightning, but you may prefer something else.

I usually carry two swords, and three bows. I just switch them out if my next target has resistance. The only possible enchantment conflict I can think of is order of activation. And that only with a elemental damage followed by Soul Trap. If your weapon strikes the target and inflicts elemental damage, then the target dies, the Soul Trap spell never takes effect. So remember to enchant Soul Trap first, then a damage spell.