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Related description from Use Magic Device:

Emulate an Ability Score: To cast a spell from a scroll, you need a high score in the appropriate ability (Intelligence for wizard spells, Wisdom for divine spells, or Charisma for sorcerer or bard spells). Your effective ability score (appropriate to the class you're emulating when you try to cast the spell from the scroll) is your Use Magic Device check result minus 15. If you already have a high enough score in the appropriate ability, you don't need to make this check.

Use a Wand: Normally, to use a wand, you must have the wand's spell on your class spell list. This use of the skill allows you to use a wand as if you had a particular spell on your class spell list. This use of the skill also applies to other spell trigger magic items, such as staffs.

Related description from Staffs:

Staffs use the wielder’s ability score and relevant feats to set the DC for saves against their spells. Unlike with other sorts of magic items, the wielder can use his caster level when activating the power of a staff if it’s higher than the caster level of the staff.

Then when a rogue with high-enough UMD skill tries to use a staff, what should be the proper way to determine the DC of that spell? As a DM, I don't think I should determine which ability should be applied to each of those staffs before generating them.

The ways I can think of:

  1. Use either INT, WIS, or CHA of that rogue, whichever is higher.
  2. Or, use INT or CHA for arcane spells, and WIS for divine spells.

And if the abilities are not meeting the requirements:

  1. If none of those abilities is at least 10 + spell level (minimum required attribute), use the minimum required attribute instead.
  2. Just use the highest abilities to decide the DC, even if it is not meeting the minimum requirement. (So a staff can become a worse option than a wand)
  3. Let the rogue make another UMD check to emulate an ability score, and use that score to determine the DC (if that score is higher than what they already have). The rogue can do this even if they have already reached the minimum requirement to have the opportunity to approach a higher DC. (Let's say somebody has INT 14 but can use UMD to emulate INT 18, then use INT 18)
  4. Just use the minimal DC regardless. When it comes to UMD, treat the staff as a wand.

I wonder if there's some kind of official Q&A to sort this mess out. If not, please let me know your opinion on how to solve this.

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By RAW a character can't emulate an ability score to use a staff, but only to use a scroll.

But if you could (spoiler: i think you should), then we need to look at how all of this works:

The steps needed to use a staff with UMD are:

  1. Emulate a class feature
  2. Emulate an ability score

First: what is a class feature?

Class Features: Special characteristics of the class. When appi- cable, this section also mentions restrictions and disadvantages of the class. Class features include some or all of the following. [...] Spells: Wizards, sorcerers, clerics, druids, and bards use spells. [...]

So casting spells is a class feature and therefore can be emulated thanks to UMD:

Emulate a Class Feature Sometimes you need to use a class feature to activate a magic item. In this case, your effective level in the emulated class equals your Use Magic Device check result minus 20. This skill does not let you actually use the class feature of another class. It just lets you activate items as if you had that class feature.[...]

Then we need to emulate an ability score, in this case it will be the key ability score of the emulated class. From UMD:

Emulate an Ability Score: To cast a spell from a scroll, you need a high score in the appropriate ability (Intelligence for wizard spells, Wisdom for divine spells, or Charisma for sorcerer or bard spells). Your effective ability score (appropriate to the class you're emulating when you try to cast the spell from the scroll) is your Use Magic Device check result minus 15. If you already have a high enough score in the appropriate ability, you don't need to make this check.

Finally, we can cast a spell from a staff without having it in our character's spell list also thanks to UMD:

Use a Wand: Normally, to use a wand, you must have the wand's spell on your class spell list. This use of the skill allows you to use a wand as if you had a particular spell on your class spell list. This use of the skill also applies to other spell trigger magic items, such as staffs

Finally, I quote again how UMD let's us emulate a score:

Emulate an Ability Score: To cast a spell from a scroll,[..]

This rule alone is enough to answer the question: we simply can't use the staff because UMD let's us emulate an ability score only to use a scroll.

...But why shouldn't a character be able to emulate an ability score to use a staff? He can already use wands(50 uses, max spell lvl 4th) and scrolls("1-3 uses", no max spell lvl) and a staff is nothing else than a themed wand withouth a max spell limit, like a scroll.

I would allow a character to try a UMD skill check to use a staff, like in your example.

To answer you: You must always emulate a spellcaster class using UMD. The caster level of the spell is the caster level of the staff or the effective one of the emulated class(UMD -20). The save DC depends on the key ability score of the emulated class: you can use your character's or try another UMD skill check to emulate an higher one.

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