As I understand it, once the Relinquished has targeted a card it can no longer be used by its original owner. And the equipped monster's original owner can't get his card back. Is that true? If it's true, the original owner won't be able to use the effects of his monster, right? What about the Relinquished's owner? Can he use the stolen monster's effect?
A: When the effect of "Relinquished" resolves, the monster targeted by it will be equipped and that monster will be placed in the Spell/Trap zone of the activator of the effect of that "Relinquished". In this case, the current controller of that card(which is now a Spell) is the activator of "Relinquished". Saying that the original owner of the monster can't get control of it again will depend. Obviously, without a card effect to do so, control will not switch over. As far as I know, there's no card effect which can 'steal' Spell cards, however, if "Relinquished" either leaves the field or gets flipped face-down, the Equipped card is destroyed and will be sent to the original owner's Graveyard. Furthermore, the original owner cannot use the effect of the Equip card, because the original owner does not control the card. In addition, all the monster effects that the equip monster has are not applicable while it is an equip card. This is why even if you equip "Gladiator Beast Laquari" with "Relinquished", "Relinquished's" attack will not be 2100, even if "Gladiator Beast Laquari" is treating its original ATK as 2100 through its monster effect. Also, as a general rule, you can only activate card effects if you control a card, unless a card effect says so, like "Extra Net". Being the original owner of a card is not relevant in the general case.
The reason why the card doesn't return to its original owner without a card effect is because "Relinquished" doesn't mention anything about only equipping for a specified time period, unlike "Junk Archer", which says that it banishes the monster until the End Phase.
A monster equipped to the Relinquished is no longer a monster but considerated an equip Spell/Trap card. Therefore, it will be returned to the hand. But whose hand?
A: The original owner's hand. If a card says return 'x' to the hand, it is always the original owner's hand, unless there is an effect like "Exchange" which explicitly mentions the other player's hand as well.
If the Relinquished is equipped by this card, will it be able to inflict piercing battle damage?
A: Yes, it gets piercing damage. This can be inferred from the text as well. It doesn't say that the equipped monster has to be equipped by "Rider of Storm Winds" in order to inflict piercing damage, nor to be able to apply the destruction substitution effect. Keep in mind that the piericing damage and destruction substitution effect are not treated as effects of "Relinquished", as any small difference might come up.
Some things to keep in mind:
The equipped 'monster' is treated as an Equip card forever. Whenever a card leaves the field(it is probably more accurate to say after), it is duplicated into a 'new' mechanical card. For anyone trying to understand rulings, this is the most important concept to grasp.
- When "Sangan" is sent from the field to the Graveyard, even as an Equip Spell card, the reason its monster effect is able to activate is because its trigger effect lays out a mechanism for detecting if it(the physical card you see) "Sangan", was sent from the field to the Graveyard(you can see this will be the case even if equipped by "Relinquished"). It is true that it was an Equip Spell card, but that effect of "Sangan" doesn't activate and doesn't trigger until it is actually sent to the Graveyard, at which point it becomes a monster again.
Before reading "2", please realize that "Reborn Tengu" used to not say explicitly 'face-up', and the rulings for "Reborn Tengu" were still the same; you couldn't activate its effect if removed from the field while face-down. You may think this explanation might be useless because of the card text, but that's fallacious, as if you do not understand "2", you will not understand the difference between "Archlord Kristya" and "Crystal Beast" rulings(i.e. no matter how much card text will not explain why rulings are true, they will just tell you what to do it and when, this is synonymous to math students being apple to apply complex formulas to problems but not understanding or being able to prove those formulas, unlike actual mathematicians who don't need problems in cookie-cutter format in order to be solved..
- When "Reborn Tengu" leaves the field, even as an Equip Spell card, the reason its monster effect is able to activate is because its trigger effect lays out a mechanism for detecting if it(the physical card you see) "Reborn Tengu", left the field. Now this is where all the scoffers start saying that I would be taking this rulings matter very far, but "Reborn Tengu" rulings will prove them incorrect, as you are not able to activate "Reborn Tengu's" effect if it was sent from the field(in face-down / set position, even as an Equip Spell card) to the Graveyard. This is yet another reason why understanding what I said above is very important. When "Reborn Tengu" leaves the field in face-down defense position, for example), it is removed from the field face-down, which means that the trigger effect "If this face-up card leaves the field" will not trigger, because that trigger effect will trigger "in limbo"(unofficial term to describe the medium of transit between any two locations in the YGO card game), and since face-down cards have no properties, there's no mechanism for which it can trigger, which is why the effect will not activate. Now at this point you may be wondering why "Sangan" can activate if sent from the field to the Graveyard, even if it was face-down either as a monster or as an Equip Spell. Again, card text will not come to the rescue for you here; you need to understand the rules or you can at least make up models that make sense. It is because its trigger effect requires that it know whether or not it got sent from the field to the Graveyard, and since "Sangan" is public knowledge while in the Graveyard, its effect can activate.
Finally, to explain what happens when you have monsters which activate when removed from the field like "Reborn Tengu" vs effects that apply(do not activate) when removed from the field(like "Grandsoil the Elemental Lord"). Even though both "Reborn Tengu" and "Grandsoil the Elemental Lord" trigger their effects at the same time(immediately after leaving the field), "Reborn Tengu" cannot activate its effect until the next chain can be built, while "Grandsoil the Elemental Lord" applies its effect as soon as possible( i.e. immediately after it leaves the field, before it even goes to its final destination). Thus, if they were equipped on "Relinquished" and "Back to Square One" returned both to the top of the deck when removed from the field face-up, which effects can activate/apply?
A: "Reborn Tengu" will not activate, because it was returned to the deck, and "Reborn Tengu" does not have an innate trigger which detects what happens if it is returned to the deck, so even though its effect triggered, by the time it would've been able to activate, it cannot. "Grandsoil the Elemental Lord" will apply its effects and skip the next Battle Phase, because as I said it applies its effect immediately upon leaving the field, and because it left the field face-up, its effect will trigger, because it provides it a mechanism for detecting if it 'left the field'.
If "Skill Drain" is applying, and "Relinquished" has "Grandsoil, the Elemental Lord" or even "Archlord Kristya" eqipped, will their effects apply if they are destroyed by "Mysical Space Typhoon"?
A: Yes, because they are not monster cards until they reach their destination to which they would be sent to. And even if they were treated as monsters, they left the field anyway so they'd be outside the range of "Skill Drain's" negation.
If "Spell Canceller" is applying, and "Relinquished" has "Grandsoil, the Elemental Lord" or even "Archlord Kristya" equipped will their effects apply if they are destroyed?
A: No, their effects will be negated, because they are Spell Cards until they reach their final destination.
Please note that the "Archlord Kristya" v. "Skill Drain" ruling on the English wiki page is actually incorrect. If "Skill Drain" is applying and "Archlord Kristya" is destroyed in the monster zone, it still gets returned to the deck, because as I hinted many times that effect applies outside the negation range of "Skill Drain", unlike "Crystal Beasts" which apply their effects while still on the field(their effects would be negated). I am not sure how long it has been incorrect but on Jwiki, the ruling for "Archlord Kristya" and "Skill Drain" was last updated 2017 April 6th.
You can check all the updated rulings on Jwiki and the official card database, or on ygo organization database page, which has many rulings translated to their English counterparts. However, what I laid forth previously is just a model to explain why the rulings are true, anyone is free to adopt another model if it makes sense in your head. PM me if you still have trouble on Pojo, username: The Chief.