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.

The second ability doesn't wear off. If you use it, Mizzium Transreliquat becomes a copy of target artifact permanently and gains the ability. It no longer has the 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 the battlefield 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.

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 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.