Yes he would be able to craft it, if none of the special abilities require a caster level higher than 5. It should not be an issue because 5th level characters are not expected to have that much money.
Given that it costs about 15,000 gold more than what a 5th level character is expected "powerful" is if the GM is allowing it. Basically if it were actually happening, there are other issues occurring in the game which have nothing to do with RAW.
The item's CL is the highest of:
- 3 (ordinary +1 bonus)
- 10 (flaming)
- 8 (frost)
- 8 (shock)
- 5 (thundering)
So the CL is 10. This doesn't matter for the crafter though. There's no minimum caster level on the requirements line for any of them. So even a first level character could make one, if they somehow had 25,000 gp, 25 days and the Craft feat (the last of which shouldn't be possible though).
What does matter is the four spells it'll take to make. For each of those you're missing, it's +5 DC. At 5th level, a wizard couldn't have the appropriate spell for the icy property, so the final DC would be 20, which is pretty easy to make. (As mentioned, they shouldn't have that much money though.)
Actually when you craft weapons the caster level is a requirement.
Creating a magic weapon has a special prerequisite: The creator's caster level must be at least three times the enhancement bonus of the weapon. If an item has both an enhancement bonus and a special ability, the higher of the two caster level requirements must be met. A magic weapon must have at least a +1 enhancement bonus to have any melee or ranged special weapon abilities.
Actually when you craft weapons the caster level is a requirement.
Creating a magic weapon has a special prerequisite: The creator's caster level must be at least three times the enhancement bonus of the weapon. If an item has both an enhancement bonus and a special ability, the higher of the two caster level requirements must be met. A magic weapon must have at least a +1 enhancement bonus to have any melee or ranged special weapon abilities.
This is a grey area and has been debated widely on the boards over the years. The wording of the magic item creation rules could be tightened up/clarified but other than occasional comments by devs the grey areas have yet to be definitively un-greyed.
So it can be as valid to say the level x3 is just another prerequisite that can be circumvented with a +5 to the DC as it is valid to interpret it as you must be at least level x 3.
For me the key things that limits player crafting are time and money. Sticking roughly to wealth by level keeps things in line, just as making adventures move along at a pace where they don't have time, assuming they have the money, to go overboard with crafting.
See Spell Storing for a weapon property that has Caster Level as a specific requirement for crafting.
The wording in that section is inconsistent. The devs have mentioned that it has slipped through multiple editions without being corrected properly.
According to the dev posts, it seems like the CL for magic weapon bonuses is no different than the CL for special properties and is not a requirement at all. Still, it's debatable, but the existence of things like the spell storing property lead me to believe that a level 5 character can make a +2 sword without the DC increase.
Edit: For the next update, maybe they can just replace Item's Caster Level with just Item's Level. It'd avoid a lot of confusion.