I asked something similar and didn't get the clarity I was seeking, so I will be more concise. If I craft a 1st level Pearl of Power at CL 1, and it gets hit with a targeted Dispel Magic, what is it's caster level for the check? Is it 1, as that was the chosen CL at time of crafting, or is it 17, the listed CL for a Pearl of Power of any level?
You use the Caster Level you used when crafting the item.
The Caster Level of an individual item is whatever you set it as, within the limits of what's possible. The Caster Levels given in the item descriptions are just the standard values. If you change one of those values in a crafted item, you use the changed value for all relevant purposes.
The pearl of power has the caster level its creator picked
The pearl of power has two construction requirements: the feat Craft Wondrous Item and be able to cast spells of the spell level to be recalled. Neither construction requirement mandates a specific caster level. (Don't worry! I'll address the FAQ shortly.) So when the creator sits down to make a pearl of power, he simply picks the caster level at which to make the pearl. Magic Item Creation says
At the end of this process, the spellcaster must make a single skill check (usually Spellcraft, but sometimes another skill) to finish the item. If an item type has multiple possible skills, you choose which skill to make the check with. The DC to create a magic item is 5 + the caster level for the item. Failing this check means that the item does not function and the materials and time are wasted. Failing this check by 5 or more results in a cursed item.
Emphasis mine. Usually (like in the case of, for example, the handy haversack) that caster level is the minimum needed to cast the spells that go into the item's creation. Magic Item Creation also says
A creator can create an item at a lower caster level than her own, but never lower than the minimum level needed to cast the needed spell.1
Note that no restriction is placed on creating an item at a higher caster level than the creator's own caster level. (The FAQ addresses this; see below.) A creator can also choose to leave one or more non-item creation feat construction requirements unmet and still attempt the item's creation. Magic Item Creation says
Note that all items have prerequisites in their descriptions. These prerequisites must be met for the item to be created. Most of the time, they take the form of spells that must be known by the item's creator (although access through another magic item or spellcaster is allowed). The DC to create a magic item increases by 5 for each prerequisite the caster does not meet. The only exception to this is the requisite item creation feat, which is mandatory. In addition, you cannot create potions, spell-trigger, or spell-completion magic items without meeting its prerequisites.
Emphasis mine. Put this all together, and, for example, a level 1 wizard can make a pearl of power (9th) that has a caster level of 1 by making a Spellcraft skill check (5 base +1 caster level minimum +5 to leave unmet the construction requirement be able to cast spells of the spell level to be recalled = DC 11).2,3
This is why, for example, a creator can't typically make a robe of gates at a caster level lower than 17 (because that's the typical caster level necessary for the spell gate even if that construction requirement goes unmet) but could make a dwarven thrower at caster level 6 instead of caster level 10 (by leaving unmet the construction requirement creator must be a dwarf of at least 10th level but caster level 6 being the minimum to grant the +2 enhancement bonus, and which can't be waived because that's not a formal construction requirement; such a creator would also have to actually be a 6th-level caster).
The FAQ dislikes, for example, caster level 1 pearls of power (9th)
The official Pathfinder FAQ (often used in lieu of errata) contains this exchange:
Pearl of Power: What is the caster level required to create this item?
Though the listed Caster Level for a pearl of power is 17th, that caster level is not part of the Requirements listing for that item. Therefore, the only caster level requirement for a pearl of power is the character has to be able to cast spells of the desired level.
Emphasis mine, and, while I can't be sure, I think there should be quotation marks around caster level in the emphasized text. That is, that's not actually a pearl of power construction requirement; be able to cast spells of the spell level to be recalled, however, is a construction requirement and that can remain unmet by increasing by +5 the Spellcraft skill check needed to create the item. The FAQ continues, saying
However, it makes sense that the minimum caster level of the pearl is the minimum caster level necessary to cast spells of that level--it would be strange for a 2nd-level pearl to be CL 1st.
For example, a 3rd-level wizard with Craft Wondrous Item can create a 1st-level pearl, with a minimum caster level of 1. He can set the caster level to whatever he wants (assuming he can meet the crafting DC),4 though the pearl's caster level has no effect on its powers (other than its ability to resist dispel magic). If he wants to make a 2nd-level pearl, the caster level has to be at least 3, as wizards can't cast 2nd-level spells until they reach character level 3. He can even try to make a 3rd-level pearl, though the minimum caster level is 5, and he adds +5 to the DC because he doesn't meet the "able to cast 3rd-level spells" requirement.
So while it makes sense, this isn't actually changing anything. The FAQ recommends, instead of a blanket level 17 on all pearls of power, that pearls of power have caster levels of at least 1, 3, 5, 7, 9, 11, 13, 15, and 17 for 1st-, 2nd-, 3rd-, 4th-, 5th-, 6th-, 7th-, 8th-, and 9th-level spells, respectively, but you, Mr. I'm-Makin'-My-Own-Pearls-in-My-Basement, can totally not do that and go ahead and make cut-rate-yet-still-just-as-expensive, easily dispelled caster level 1 pearls of power (9th) if you want.
"Then a creator that can make an arbitrarily high Spellcraft skill check can make a magic item with an arbitrarily high caster level?"
Yes. The FAQ says so. This is of little help with items that PCs get the most say in (i.e., for example, potions, scrolls, staffs, and wands, all of which must have their construction requirements met and all of which put a premium on caster level), but for, for example, magic rings, wondrous items, and magic armor, shields, and weapons, the creator can set the DC for the caster level that's part of the Spellcraft skill check to whatever the creator wants as long as its at least equal to the minimum necessary caster level for casting the spells in the item's construction requirements (and whatever's mandated by the item creation feat, obviously).
Certainly, there are some shenanigans that can be pulled by setting an arbitrarily high caster level for magic rings, wondrous items, and magic arms and armor, but magic items that deviate from the game's norms (and, really, any magic item created by a PC) must be approved by the GM, and the GM is supposed to shut down a creator that proposes a crazily unbalanced custom item. Just because something can be done doesn't mean the GM must allow it to be done.
1 Former Pathfinder developer Sean K. Reynolds seems to forget about this requirement for creating magic items in this otherwise informative 2010 Paizo messageboards thread.
2 We won't worry about precisely where that level 1 wizard gets the 40,500 gp in raw materials necessary to create that pearl of power (9th).
3 Likewise, this is why to create a handy haversack the level 3 caster makes a Spellcraft skill check (5 base +9 minimum typical caster level for secret chest +5 to leave unmet the construction requirement secret chest = DC 19).
4 A previous version of this question asked if a magic item's creator could set the caster level of an item at pretty much anything as long as it was at least or higher than the minimum caster level. This emphasized text in the FAQ says Yes. Further, a previous version of this question asked if doing so would count as an unmet magic item creation requirement therefore increasing by +5 the Spellcraft check DC necessary for making the item. The game says usually No as actual caster level is only rarely a formal construction requirement.
Short answer, If you make it CL 1, it has CL 1 for the dispel check.
See this faq for more details, but it's really very simple.