AP connects with a Thief of Sanity and chooses to exile a Rix Maadi Reveler from NAP's library. Can the Reveler be cast for its Spectacle cost, and can mana of any color be used to pay for the black and red mana in its cost?
AP casts Hostage Taker and exiles NAP's Venerated Loxodon. Can AP cast the Loxodon using Convoke? If yes, what combinations of creatures and mana can they use?
Effects like this are interesting to me because of how they've evolved over the years, and the rules implications therein. Older cards would let you cast spells without paying their mana cost. Think Cascade. Casting a spell without paying its mana cost is a subset of alternative costs, rule 117.9:
Some spells have alternative costs. An alternative cost is a cost listed in a spell’s text, or applied to it from another effect, that its controller may pay rather than paying the spell’s mana cost. Alternative costs are usually phrased, “You may [action] rather than pay [this object’s] mana cost,” or “You may cast [this object] without paying its mana cost.” Note that some alternative costs are listed in keywords; see rule 702.
The mana cost of a spell is whatever is in the top right corner of the spell, just as its Converted Mana Cost is the generic sum of its mana cost. Casting a spell without paying its mana cost means not paying for what's in the top right corner, essentially turning it into a 0. You'll still have to pay for any additional costs, whether they are imposed by the card itself or by other effects. If you Cascade into a Harrow, you'll still need to sacrifice a land in order to cast it. If you Cascade into a Lighting Bolt, you must pay 1 for Thalia, Guardian of Thraben. While this might feel like a negative, it also gives you the ability to pay for optional additional costs like Kicker.
Because "casting a spell without paying its mana cost" is an alternative cost, you cannot then choose to cast the spell via some other alternative cost. If you Cascade into Nourishing Shoal, you can only choose to cast it for X=0, and you do not have the option of trying to exile a green card in order to get an X-value. Another good example of conflicting alternate costs is when you give a spell flashback with Snapcaster Mage. You can only play the spell via its flashback cost. You can't pitch for a Force of Will, for example.
So what of Thief of Sanity and Hostage Taker? Neither of these cards offers an alternate cost to cast the spells. You are still on the hook for paying a cost of some kind. For the vast majority of cards, that means paying their mana cost. However, it does mean that you can cast these spells for alternative costs. You can pay a Spectacle cost (if you've met the damage criteria), and the mana clause will apply as well, so you can pay for Rix Maadi Reveler's Spectacle with all blue mana if you want.
Convoking a Venerated Loxodon yields one slightly different result. Since you are casting it, you can use Convoke to help pay for the spell, but the color-filtering only applies to mana spent to cast the spell. You cannot Convoke with a non-white creature for that portion of the spell. You will either need a white creature to Convoke with, or you will have to pay that portion with mana, which can be any color.
Both of these are good examples of how getting something for free comes with some opportunity costs. By being "forced" to pay for these spells, you can get the additional bonus of the Spectacle trigger upgrade on Rix Maadi Reveler, or the counters on creatures that help Convoke your Venerated Loxodon.
When you are evaluating effects like this, remember that "casting a spell without paying its mana cost" is an alternative cost, and will preclude any other alternative costs like Spectacle and Madness, and any associated benefits from casting spells that way. If you are just asked to cast the spell (under normal circumstances to spend mana on it) you can instead apply an alternative cost, which includes casting variants like Spectacle and Evoke, or mana free options like on Force of Will or Archive Trap.
Addendum: While writing this post, another interesting card with this type of effect was spoiled, God-Eternal Kefnet.
You can see that this follows in the footsteps of Thief of Sanity and Hostage Taker in that you are casting the spell. It's a little different because you are casting a copy of a card. But if you've met the casting condition for things like Spectacle, you can cast this via alternative costs. Whatever alternative cost you choose will get the benefit of the cost reduction.
One of the more common questions about effects like this is "Can you just float this copy in limbo until you need it? It doesn't give a duration for when you can cast it." The latter part is true enough. Thief of Sanity and Hostage Taker tell us "for as long as it remains exile." Kefnet doesn't. In the latter cases, when no duration is listed, the card can only be cast immediately during the resolution of the triggered ability. And yes, you can cast a sorcery even tough it isn't your main phase. Why? Because the trigger lets you. Technically, you couldn't even cast an instant at this time (during the resolution of a triggered ability) if the ability did not let you.
Finally, I've seen questions about whether you can Miracle cast this copy. Seems possible because Miracle is an alternative cost, right? The problem here is the part about fulfilling the casting conditions. Just like you would need to have damaged your opponent to be able to Spectacle a copy off of Kefnet, you need to meet the requirements for Miracle, which are a bit more complicated. In fact, the only way to cast something for its Miracle cost is during the resolution of its own trigger. If you reveal a Miracle card for both Kefnet and its own Miracle ability, you could cast the original card for the Miracle cost and the copy for its regular mana cost minus 2.
