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The one who mastered battle is the arena's Grand Champion, and Cicero notes that he killed said champion.
The specific relative is the Night Mother, and she represents the DB, and the citizens aren't exactly appreciative of them.
To the listener, not Cicero, she gives her orders,
But Cicero loves death and blood, and also wears a jester's clothes.

It is not a living thing.

It is not a quest, but the name of a quest is alluded to in the riddle..

Starting with the fact that the riddle alludes to a quest (and assuming that the quest it alludes to is related to the actual answer), the first quests that pop into my head are The Man Who Cried Wolf,(A man pretending that something's wrong) and The Throat of The World (A splendid peak to you must belong,).

The first two lines indicate that something changes depending on some circumstance (level, number of shouts learned, quest decisions etc,.). The third line indicates somebody is lying (notable liars include Ancano, Beem-Ja, Jaree-Ra, Telrav). The fourth line indicates a mountain is involved. The fifth line is more tricky, either it involves making somewhere between 5-7 people happy, or it involves doing between 5-7 quests. The sixth line references something that comes in groups of three; things like dragon shouts, the Nightingale trinity, and Aetherium items come to mind.

Finally, we know it's not a living thing (although it may be a zombie of some sort ) or a quest, so it's likely an item (more specifically a quest item) possibly even a book.

One line in the riddle is a quest involved with the answer. One of the quests you mentioned is the correct one.
(Side quests may also be referenced).

You are right about the first two lines.
The forth line assumption is incorrect.
You are right about the fifth line.
The sixth line does not indicate something that comes in groups of three.

It is not a zombie. *wink wink*

You never know what you may get,
Until conditions are rightly met, The rewarded weapon is levelled

A man pretending that something's wrong, One of the prequisite quests is "The Man Who Cried Wolf"
A splendid peak to you must belong, ???
Near half a dozen must be pleased, 5 "assist the people" quests are needed
One type of three can then be seized. ???

A splendid peak to you must belong, This line is a play on words, 'splendid' and 'peak' being the synonyms (i admit they are not the best though- was tricky to get exact ones) and 'belong' implies that it is officially yours.

You never know what you may get - Skyrim rewards system.
Until conditions are rightly met - To become a thane you must do:
A man pretending that something's wrong - The Man Who Cried Wolf.
A splendid peak to you must belong - Proudspire Manor.
Near half a dozen must be pleased - Help 5 people.
One type of three can then be seized - Reward can be sword, greatsword, or mace.

and no, I'm not going to make a riddle so you can pass it on to next one. My Pegasos is limping so badly.