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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The second ability doesn't wear off. If you use it, Mizzium Transreliquat becomes a copy of target artifact permanently and gains the {1}{U}{R} ability. It no longer has the {3} ability (unless it copied itself or another Mizzium Transreliquat).

An artifact's "copiable values" are those printed on it, as modified by other copy effects, plus any values set by "enters as" abilities. Counters and other effects aren't copied. For example, if you copy a Gruul War Plow that's been turned into an artifact creature, the result will be just an artifact.

If you use Mizzium Transreliquat's abilities multiple times in a turn in response to one another, then each time one of those abilities resolves, it will overwrite whatever the permanent was copying. The Transreliquat will wind up as a copy of the artifact targeted by the last ability to resolve. When the turn ends, all instances of its first ability will wear off at the same time. If one of those was the last copy ability to resolve, the Transreliquat will become what it was before those abilities resolved. This will likely be either the original Mizzium Transreliquat or whatever it copied with the last instance of its second ability to resolve.

The results of all copy effects are copied. If Mizzium Transreliquat is copied, the copy will be a Mizzium Transreliquat after applying all copy effects currently affecting the original. For example, Copy Artifact copying a Transreliquat that's using its first ability to copy an Izzet Signet will be an Izzet Signet. The effect won't wear off at the end of the turn; rather, the Copy Artifact will remain an Izzet Signet for the rest of the game.

If you use the first ability, Mizzium Transreliquat loses both of its copy abilities until the effect wears off (unless it copied itself or another Mizzium Transreliquat). When it wears off, Mizzium Transreliquat loses all abilities it gained this way and goes back to being what it was before.