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Similar questions have been asked over the years, and I think I have a handle on this, but wanted my math double-checked.
My logic seems to hinge on the difference of a bonus to the Skill vs a bonus to the Check. Here is the text of the ability:
"Versatile Dance (Ex)
At 2nd level, a dervish dancer gains a bonus equal to half his level on Perform (dance) checks. He can use his bonus for his Perform (dance) skill in place of his bonus for Acrobatics. "
So, for my example
A 2nd level Dervish Dancer bard using Versatile dance to use Perform (Dance) on an Acrobatics (move through) check. (Has levels in other classes)
On the perform side
Skill points: 4
Class Skill: 3
Cha mod: 3
1/2 class level from Versatile Dance: 1 untyped
Masterwork Tool (Hammer Pants): 2 circumstance
Cracked Pink & Green Ioun Stone: 1 untyped
Does not apply
Bracers of the Glib Entertainer: 5 competance (applied to perform CHECK, not Skill)
Total: +14
On the Acrobatics side
Belt of Tumbling: 4 competance (to the Check)
acrobatic feat: 2 untyped (to the Check)
Armor check penalty: -1 (to the Check)
Does not apply
Skill points, class skill, Dex mod, masterwork tool (shoes?)
Cracked Deep Red Ioun Stone: 1 untyped (to the skill)
oddly, would apply if we didn't have the belt.
Cracked Vermillion Rhomboid Ioun Stone: 1 competance (to the check)
Total: +5
Total bonus: +19. Is that correct?
Interesting questions with no items to support them for now:
Would an item with a competance bonus to perform (skill) stack with competance bonus to acrobatics (check)?
I suppose other mod types (racial, morale, insight, dodge?) would also fall on either the skill or the check.
You use your bonus to perform instead of the bonus to Acrobatics in all cases. You can choose to roll Acrobatics instead if you like.
Anything that gives you a bonus to perform adds to your versatile perform. Anything that gives you a bonus to acrobatics adds to your acrobatics. This means that you use perform to jump, tumble, balance, etc. and can get +5 from the bracers of the entertainer, can get +3 from a circlet of persuasion, can take skill focus (perform) or prodigy to up your checks.
This also means that if you gain bonuses to acrobatics they do not apply to your check if you use perform. Vice versa with perform and acrobatics. If you get +20 to Perform (dance) and roll acrobatics, you don't get the +20; same with acrobatics and perform. So, if your movement speed goes up 30ft by, say, haste, you don't get +12 to jump checks if you use perform instead. That's the only complicated bit about it.
Dividing 'skill check' and 'skill bonus' into two categories is silly enough that you might as well divide 'hp' and 'hit points' into two columns.
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That contradicts what Jacobs said back in 2010.
"But for magic items, I would rule that something like boots of elvenkind WOULD grant you the bonus even if you were using versatile performance by using your Perform skill. The boots are supposed to make your Acrobatics work better, and it makes no sense at all for a character specialized in being all Acrobatic to suffer just because the rules allow her to use a different skill check.
Versatile performance is supposed to widen a bard's skill selection, in other words, not to have stealth punishments that prevent you from using items."