The Pathfinder Core Rulebook FAQ contains this exchange:
Does using a potion, scroll, staff, or wand count as "casting a spell" for purposes of feats and special abilities like Augment Summoning, Spell Focus, an evoker's ability to do extra damage with evocation spells, bloodline abilities, and so on?
No. Unless they specifically state otherwise, feats and abilities that modify spells you cast only affect actual spellcasting, not using magic items that emulate spellcasting or work like spellcasting.
This means, for example, a creature that activates a wand of dimension door—despite Wands on Activation saying that a creature using a wand "cast[s] a spell from a wand"—will not realize the benefit of that creature's Dimensional Dervish feat. A creature that possesses the feat Dimensional Dervish takes a standard action (not a swift action) to activate a wand of dimension door just like almost everybody else.
Likewise, the feats Dimensional Agility and Dimensional Assault also specify actually casting the spell dimension door so their benefits won't be realized upon the creature activating a wand of dimension door either. And the feat Dimensional Savant is contingent upon the creature realizing the benefit of the feat Dimensional Dervish, so realizing the benefit of the feat Dimensional Savant also isn't possible if such an attempt is made to realize that feat's benefit via a wand of dimension door.
However, the ring of spell storing itself specifically "contains up to 5 levels of spells… that the wearer can cast." This might be enough for a GM to allow a dimension door spell stored in ring to count as the wearer casting it himself. At 50,000 gp per ring—each of which can hold but a lone dimension door spell and each of which requires a new dimension door spell to be cast into it after the present dimension door spell's been used—, it's unlikely a PC will find this a good workaround, but this GM would likely allow the ring's stored dimension door spell to count as the wearer actually casting the spell for the purposes of feats like Dimensional Dervish. Ask your GM.
Finally, a ring of wizardry IV merely doubles the wearer's number of 4th-level spell slots in which the wearer can prepare spells. Subsequently, the ring's wearer still must really cast those spells, so feats like Dimensional Dervish apply when casting a dimension door spell prepared in one of those ring-granted extra slots. And, like the ring of wizardry, a pearl of power "enables the possessor to [take a standard action to] recall any one spell that she had prepared and then cast that day" and, subsequently—usually next turn or later—, cast that spell normally. Thus if the spell recalled using the pearl is dimension door, the caster can, after using the appropriate pearl of power, cast the spell and benefit from feats like Dimensional Dervish normally.
It's normally impossible to realize the benefit of the feat Dimensional Dervish with the spell dimension door if the dimension door spell's cast as a standard action. The feat's benefit says, "You can take a full-attack action, activating abundant step or casting dimension door as a swift action." Those conditions are necessary to realize the remainder of the feat's benefit, the game indicating those requirements with the phrase If you do….
Note: According to FAQ the feat Augment Summoning when used in conjunction with spell-like abilities is specifically an exception to the FAQ entry this answer quotes. Nobody's entirely sure why Augment Summoning gets special treatment, and the FAQ's ruling on that feat's interaction with spell-like abilities shouldn't be taken as precedent and applied generally.
