A few questions on Queen Barenziah.

She is a dumner, right?
Is she still alive?
Some sites say that she is a mage, is this true?
How come we haven't gotten any references about her in Skyrim, other than seeing/obtaining her crown?
What would her build be? (Battlemage, Spellsword, Pure Mage, Mage Thief)

She's long dead and she was a dunmer. There are a couple of books in the TES world about her.
Edit: http://www.uesp.net/wiki/Lore:Barenziah

There are also some hints that Karliah is the daughter of Barenziah and Jagar Tharn.
Barenziah is likely dead but you never know with her. She hasnt been confirmed to be dead so who knows, she might pop up again.

You sure she isn't dead? Hmmm i thought she was...

Her crown was looted and scattered all over Skyrim, which makes me think she probably is dead. Though, if you think about it, why Skyrim, of all places?

It's possible that she fled Mournhold when it was attacked (no doubt evacuating the royal family was given the highest priority), and then crossed the border into Skyrim, where she hocked her crown and travelled around incognito.

But it's not like she's ever done that before.

She'd be well over 600 old years by the time of TES V. Sure, some Dunmer can live that long, but she was considered to be an old woman in TES III.

According to Vex, when you bring her all 24 stones. the stones havent been togeather for the last two eras. Her crown had to have been looted long before the destruction of Morrowind.

Dovahkiin: I've found all the Stones of Barenziah.
Vex: Unbelievable! You must be the first person to set eyes on all twenty-four of them in... well, the last two eras.