We don't know how Mab's intelligence network works.  Specifically, we don't know how Mab became aware of Lea's involvement with Margaret LeFay and her son.  We only know that at some point Mab became interested in seeing Harry in action, to make her own judgement about his potential.   

It might have been Summer Knight when Mab took on Harry's "contract" with Lea.  The three favors he owed her for the "confidence" she gave him when he was sixteen to take on Justin.  Mab knew that what Lea did was mostly song and dance and that it is no small thing that a sixteen year old apprentice can take on, defeat and kill a fully trained warden of the White Council.

Well, in Battle Ground, Murphy is able to do the things she does because of the magical boost she's getting from Mab. Bob explains it to Harry and even holds up Murphy as an example of how it's working. If Murphy had survived Rudolph, I suspect she would have been left in charge to hold the impromptu fortress while Sanya, Butters and Harry went to ambush the Formor.

That was before the boost. 

She was an extremely-advanced martial-arts practitioner; many of them are startlingly-competent.

  She had limits, Nic wiped the curb with her.. Then she removes her own casts, one minute she can barely walk, the next she manages to surprise and take down a Valkyrie? 

I personally knew a "Murphy"-like person -- cute, small, blonde, senior aikidoka.  Nice lady, and when she came at you seriously she was the scariest person I've ever met (this with years of karate behind me).  I have literally seen another black-belt take a glance at her then turn and run away, on the mat!

Maybe when she was physically whole and in shape, but as of Peace Talks and Battle Ground Murphy wasn't.  Even if she were in splints and doing physical therapy all along after Nic got done with her... She apparently wasn't because she had to cut her cast off, if you have ever worn a cast for a time or recovered in splints after orthopedic surgery, you know the limb doesn't instantly recover, even if you are doing physical therapy all along..

When Murphy added the Sword at Chichén Itzá... yes, that turned her dial up to "11," and I liked it -- for her character, for the story -- that she decided to put the Sword back down again, afterwards.  The post-Changes Murphy was back to being a mortal, not Supermurphy; but she was much more being stretched to her limits, and barely-keeping-up.  Before that, with a couple of exceptions (vs the Nightmare & vs. Papa Raith), Murphy was mostly on top of even the supernatural situations.

Which just wasn't realistic even for a fantasy.

From the high-fantasy action-adventure aspect of the Casefiles, Murphy was no longer a "viable" adventuring character to accompany Harry.

Which Murphy should never have been to begin with in my opinion. When we first meet her, smart, strong,a good shot and martial artist, she didn't need to become an action-adventure figure to be compelling.  Michael doesn't need any super human abilities, he had a Holy Sword, but he does pretty good without that.

Actually when you think of it, Murphy's actions was really a form of suicide.  With her injuries she wasn't up to really fighting any pitched physical battles, and actually could have made things worse had Harry been distracted trying to protect her.  Yes, she caught a bullet from a paranoid cop and died, and did manage to take out a giant with a bazooka before she died, but in the later physical battles where even the likes of Odin and Mab barely made it, she wouldn't have survived.  Yes, hard to stay behind at Mac's place to protect the people there, but Harry wasn't being over protective of her in doing that.  Actually it was her ego once again that she knew better, so she failed to follow orders, that's what got her killed.

I think Harry&Elaine were (largely) being taught together.
They certainly shared a lot of homework assignments!

I'm sure there were some lessons specific to each of them, while the other did bookwork &c.  Their respective strengths and weaknesses differ, after all (not that we know a lot about Elaine's, but we see their preferred methods aren't identical).
 
I presume Elaine got largely-similar treatment to what Harry got; if Justin was notably harder or easier on her, we'd have heard about it by this time:  it's the sort of thing kids remember even into adulthood.

However at this point, we don't know..  The subjects taught might have been the same so that homework assignments can be worked on together.. However if the slant on that same subject is a bit different, the outcome can be different.  Also we presume it was magical homework they shared, it could have been a math assignment from school.  For some reason Justin attempted to enthrall Elaine first, if she was in fact enthralled. Target of opportunity? As he taught them did he find her more vulnerable?  Here is another thing that really doesn't add up.. Supposedly Elaine stayed home from school that day because she was sick.. Harry cut class to return home to be with her..  Where did Harry think Justin would be during this?  Did Justin have some kind of day job? Was Justin so involved with his magical research that he wouldn't notice that Harry cut school? Did Justin even care if Harry cut school?  Apparently it was no surprise that Harry would cut school, Justin and "enthralled Elaine" were waiting for him with a straight jacket..  The point I am getting at is, Justin counted on Harry cutting class and coming home when he did, and then trapping him.  Would he have attempted it if Harry had not cut class that day? 

After reading Peace Talks, he did look for Harry and he did find him. But he didn't take him in, because he was taken care of. I think he planned on watching, if little Harry started showing magical talent at some time. If it never happened, Harry would never have known him. If Harry showed talent, Ebenezar would have taken him as apprentice or apprentice him to an other wizard he trusted.

  Oh wonderful, a fightened six year old orphan all alone in the world, ward of the state, in an orphanage... His only known living relative ignores him, but might take an interest if little Harry showed any talent as he nears the age of eleven or twelve.. This was for Harry's own good, mind you, I guess Eb could rationalize a couple of reasons for doing that... 

But then Harry was adopted by Justin. Maybe Justin even got a tip from Ebenezar. During this time Justin was still supposed to be a respected warden, so good master material for Harry. But why did Eb approve of Justin not introducing his apprentice to the council?

Better question, why is it that only a couple of trusted wizards even know that Harry is he grandson?

After Malcolm died, Lea took care that Harry disappeared from all mundane and magical screens, because of a deal Margaret made with her for Harry's protection. So nobody could find Harry. Not even powerful wizards with blood relation. Justin was just screening the orphanages for weird incidences. That's how he found Harry. It was the impossible long jump.

Is it?  I don't buy that Justin decided one day in his retirement since he had nothing better to do, that it would be great to raise a couple of kids. So he scoured all the orphanages in the country looking for kids with talent, better yet kids that could qualify as star born, to adopt. I think there is a lot more to this story that we don't know yet.

I think #2 is more likely, considering Morgan's journal entry. I guess, Justin still appeared at council meetings, but never told any other council wizard about his apprentices. But I also can totally imagine Ebenezar knowing where his grandson was, but just not interfering. Like he seemed to have done with his own daughter.
I hope this will be cleared up some day.

I hope so too, there is also the little matter of why didn't Eb reveal to young Harry when he took him in under the Doom that he was his grandfather?

Bottom line? Eb shouldn't be lecturing Harry on how to raise his own daughter!  Wizards who live in wooden houses shouldn't be throwing fire balls!

So while there may not be any active "romantic relationship," there's still a background tension there (much as there was between him and Murphy, for many books).

But that's the point, once the sexual tension is consummated it changes the relationship. It also changed Murphy, what made her interesting as a character for me was she was a strong female, was devoted to being a good policeman and the law, but was open minded about Harry's world and what he faced.  This gave her her own conflict to deal with, the sexual tension between her and Harry only added to the mix.  Actually I think it would have been better if Jim had stuck with Murphy's original reasons for not wanting a sexual relationship with Harry.  They were very good reasons, once they were abandoned, while he tried to make her more kick ass than ever, I found her a lot less interesting and a lot less believable.

Perhaps he got wind of the Leansidhe coming to check up on Harry or even take him away? Justin was a Warden, but I doubt he could have stopped her if she were determined or bound by her deal with Maggie Sr.

Or he underestimated how quick a 16 year old kid can move when he is scared.

Even if she wasn’t enthralled he was planning to act against Harry.  Why… then?  Why the clumsy action with poor preparation?

Over confident perhaps?