Elaine is the wizard the White Council should really be afraid of.

[KurtinStGeorge]

Just to nail down one small point.  To a fairly high degree, we can be certain that no one else on the Council knows that the Gate Keeper has the means to detect Nemesis.  From Cold Days:

Harry -  "Steel," I said.
Rashid - "Pardon?" he asked.
Harry  - "Your, uh, other eye. It was steel before."
Rashid - "I'm sure it looked like steel," he said. "The disguise is necessary when I'm not here."
Harry  - "Your job is so secret, your false eye gets a disguise?" I asked. "Guess I see why you miss Council meetings."

When he is playing his role as a member of the Senior Council, the Gate Keeper does not share much information, unless he feels the need to do so.

[Mira]

Just to nail down one small point.  To a fairly high degree, we can be certain that no one else on the Council knows that the Gate Keeper has the means to detect Nemesis.  From Cold Days:

Harry -  "Steel," I said.
Rashid - "Pardon?" he asked.
Harry  - "Your, uh, other eye. It was steel before."
Rashid - "I'm sure it looked like steel," he said. "The disguise is necessary when I'm not here."
Harry  - "Your job is so secret, your false eye gets a disguise?" I asked. "Guess I see why you miss Council meetings."

When he is playing his role as a member of the Senior Council, the Gate Keeper does not share much information, unless he feels the need to do so.

Exactly, and the same goes for Harry.. Rashid also confirmed that. page 341 Cold Days

" I will do what I can.  If we both survive the next several hours, I will settle matters between you and the Council, which knows only as much about our roles as it needs to--and that isn't much.

They didn't test her in the DuMorne aftermath, no (when they collected Harry); they never connected her with DuMorne; she only appeared on-scene much later.

Warden Ramirez tested Elaine, out in LA.  Harry called to "check up on her bona fides" & Carlos confirmed it.

She wasn't quite up to WC standards... but she was close, and recognizably well-trained in WC fundamentals.

He joked that if the Rampire War continued its horrible attrition, they might have to lower their standards a little bit -- just to keep WC numbers up -- and she'd be in.

But she faked it so well, she fell off the radar.