Harry did, he rationalized that he wasn't doing anything wrong technically because Sue wasn't human... However did the White Council have a better answer for Cowl and Company? They didn't seem to did they.

Maybe, but they kept him on as a Warden for some time after that...

Politics -- and fear of being murdered by Rampires -- make strange bedfellows.
They needed him.

And maybe there were people arguing "keep your friends close, and your enemies closer!" and suggesting the White Council would do better to bring Harry into the wardens, where many other wardens would be routinely interacting with him... all the better to spot warlock-ism's, eh?

... After one year with Lea she is a force of nature in Bombshells? Against the fomor ...

Molly was also mentally/emotionally fragile, and (as an unstable mind-mage) incredibly dangerous to her friends as well as enemies.
And she was getting worse, not better, until Harry came back.

The problem is the White Council has selective memory where Harry is concerned.

No one can deny that many of Harry's exploits look shady from an outside view. But they don't give him credit for the bright ones or the times that he does the right thing when it might have been wiser to step aside.

Yes, he started the war with the Red Court (Though it was more that he gave them a pretext to put their plans in motion.), but he did it to save innocents from slavery and death. He resolved the war between Winter and Summer. He stood with the KotC as the Denarians tried to start anew Black Death. He stopped Kimmler's heirs. He again stood with the KotC to prevent the corruption of the Archive. He uncovered the traitor within the Council. He destroyed the Red Court. He dealt the deciding blow to Ethniu.  These are the things that they know about but give no credit for. Saving the world in Cold Days and being literally on the side of the angels (Multiple casual interactions with Archangel Uriel come to mind.) would be more to his good but they don't know about those beyond what Harry's reported.

The thing is:  it's not the White Council's job to extend trust, to look for more-charitable explanations, to hope for the best.  They do not allow themselves to mind-probe motivations & intentions.

They go by actions and likely intentions.

Their job is to find and stop sorcerors before they do too much damage.

Picking one of your examples:

Harry showed up with Sue, heroically rescued a bunch of wardens, and went on to heroically prevent any of the Kemmlerites from finishing the Darkhallow.

But of course EvilHarry^(tm) would do exactly that:  EvilHarry wouldn't want any of the other Kemmlerites to become a protogod of death-fueled vengeance.  Note that all the other Kemmlerites were also equally-avid about stoppying anyone (but themselves) from succeeding, just like Harry was!

But  Harry showed up to that party riding a 66-million-year-old necromantic monster.  You don't get that sort of power in moments (I mean, Harry literally did... but so far as the Council is concerned, you can't... so obviously Harry didn't).  What Harry did -- riding in on Sue -- proved to the council beyond all reasonable doubt that he had been secretly-studying Black Magic (to wit, Necromancy) for many years; in effect, showing that he was likely a disciple of one of Kemmler's disciples, and every bit as dangerous as any of Kemmler's disciples themselves!

... or if the writing gives that impression it becomes a matter of which do you believe?

If the writing is ambiguous, then it clearly becomes a case of "believe whatever you prefer to believe," with a side of "please don't present preferences as if they were facts" and topped with speculation as to whether Jim just got sloppy there, in his unclarity... or if the ambiguity is intentional (either he wanted to obscure the truth, or he wanted to hold his options open for future storytelling uses, or etc).

... I don't think Charity hid that from Michael ...

I am quite certain that she did.
She was desperate that Harry not tell Michael, or anyone else in her family.
Later -- driving Michael & Molly home from the warehouse where Harry managed to get the Doom levied -- Michael wondered how Molly could have wound up with magic, begins to ask about heritibility, and Harry has to fast-talk Michael to hide Charity's secret.

Now... Michael isn't a stupid man; when it comes to matters supernatural, he is in fact quite clued-in.  It's entirely-possible that he realized how important it was to Charity not to have to admit to Michael that she has been a wicked sorceress, and he allowed Charity & Harry to think he hadn't seen through the deception.  Or not.

But if he knows -- Charity isn't the one who told Michael.

... Or what if Harry has no protection now, it is because what he and Murphy felt for one another wasn't really true love? ...

We know Harry was protected during Peace Talks -- he burned Lara (surprising them both).

AFAIK, we don't know anything about after Rudy killed Murphy.
 

... With Margaret and Malcolm I think it is implied that they were protected from Lord Raith somehow, probably by true love, but again there is no clear evidence in the book. One can just assume ...

I think Margaret probably had a very good idea what sorts of magical resources Papa Raith had available; whampire's I think can have minor magical gifts, but cannot be very-potent as wizards (not nearly at the White Council level!).

One of the things Harry has stressed (repeatedly) is:  when a WC Wizard knows what they face, they can often prepare an overwhelming counter to that.

Add to that:  Margaret LaFey may have had bargains in place to prevent being tracked; if not, she likely arranged them ASAP after leaving Papa Raith!  And we know (from Turn Coat, where Morgan bargained with the fae for this) how the Fae can & will make somebody untraceable; I expect Lea (and certainly Mab!) can do the same as Morgan got from Summer.

...  Nemesis would have to do more than make Thomas believe it, it would have to be able to produce the burn when a WV tries to feed, to maintain such a deception.

This hardly seems insurmountable.

Nemesis is a Walker, one of the most-powerful of all Outsiders.

If Justine is Nemfected, it hardly seems outside Nemesis' power to create some burns when a whampire touches her.