#changing targets

Something becomes the target of a spell or ability when it goes from not being the target of that spell or ability to being a target of it. This can happen both when the spell or ability is put on the stack and when a target of that spell or ability is changed.

Note that it has to go from "not a target" to "a target" in order for ward and other "becomes a target" abilities to trigger, so either not changing the target (i.e. choosing the same target) or making it the target of a different part of the spell in addition won't cause these abilities to trigger additional times.

Venerated Rot Priest will trigger twice: Once for the original casting of the aura and once when the copy is created targeting Ivy.

That works correctly. 

You need to change the target first and then grant protection to Spellskite. This will cause the spell to not affect Spellskite (and to have no effect if it has no other targets).

Giving Spellskite protection before you try to change the target won’t work because the target won’t be able to change to Spellskite as it would be an illegal target.

When Spellskite’s ability resolves, it won’t be able to change any targets of Sylvan Reclamation to itself because it’s already a target of that spell. Sylvan Reclamation can only target the same object once because it only uses the word target once.

Kolaghan’s command is a different case since the different modes each use the word target and so Spellskite can be targeted by both the damage and artifact destruction modes of that spell. Ulamog, the Ceaseless Hunger’s trigger works the same as Sylvan Reclamation and can only target Spellskite once.

While you can always activate Spellskite’s ability targeting any spell or ability, the target of the spell or ability will only change to Spellskite if Spellskite would be a legal target for it.

When Spellskite is equipped with Lightning Greaves, it has shroud and thus can’t be the target of spells or abilities. This means that it can never be a legal target of a spell or ability and thus its ability won’t be able to change any targets in this case. You can still activate the ability all you want, though.

So first of all, yes, this can work if the spell whose target is being changed can legally target Ricochet Trap (i.e. a red instant spell on the stack).

So, let’s say you go to Lightning Bolt your opponent, but they try to counter your Lightning Bolt with Dispel. How rude... Of course you won’t stand for this and so you cast Ricochet Trap targeting Dispel. 

When Ricochet trap resolves, you get to change Dispel’s target. Changing the target isn’t optional so we have to change the target if possible. Let’s look at your options: All three spells on the stack are instants, which is a good start. Dispel is already targeting Lightning Bolt, so choosing that wouldn’t be changing the target. Dispel can’t target itself because a spell can never target itself. That only leaves Ricochet Trap which is still on the stack even though it’s in the middle of resolving right now. So you change the target to Ricochet Trap and then Ricochet Trap finishes resolving and goes to your graveyard.

When Dispel now goes to resolve, its only target is illegal (because it’s no longer there) and so Dispel is countered on resolution. Lightning Bolt then goes on to resolve as intended.

Relatively useless, yeah.

You won’t be able to change the target of any spell an opponent controls with it, since the target doesn’t change if Spellskite isn’t a legal target for the spell or ability when its own ability is resolving.

What would work giving Spellskite hexproof AFTER it has changed the target of a spell or ability to itself. This can be done with cards like Ring of Evos Isle. (It won’t be able to change any further spells or abilities to target it that turn, however.) When the spell or ability then tries to resolve, its target is now illegal and so it won’t affect Spellskite, and will be countered if Spellskite was the only target.

Yes, you can.

When you’re changing the target of Counterspell with Redirect, Redirect is still a spell on the stack at that time and thus a legal target to choose for Counterspell. (Note that you can’t choose Counterspell as a target for itself because a spell can never target itself.)

Once Redirect finishes resolving, it’s put into its owner’s graveyard as normal. Then when Counterspell goes to resolve, it checks its target and sees that the spell it’s targeting (Redirect) is no longer on the stack. Since all of its targets are now illegal, Counterspell is countered and doesn’t resolve.