Since the stated purpose of any focus is to improve precision and fine control, I would think that a focus as impressive as Little Chicago would drastically decrease Harry's sloppiness.  This is yet again an indication of Cowl's fundamental might and ability.

Harry is often nearly as sloppy in his methodology as he is in his execution.
A poorly formulated but well controlled spell remains poorly formulated.

In the novels when Harry observes Cowl using Little Chicago he gets noticed by Cowl and is attacked via magic. In Your Story on page 297 this "Extended Divination" spell is described causing "a mostly-invisible, spectral projection of the caster appearing in the location of the monitoring’s point of view". Modeling a "looking back in time" spell as an extended divination one could argue that a spectral projection would appear in the "past" and could therefore influence the past and present.

I don't think that's so much inherent to divinations in general as it is likely another instance of Harry being powerful but sloppy.  I rather suspect that it would, in fact, have been a Compel for Harry's character based on his 'magical style' aspect.  Other practitioners likely would have executed an equivalent spell substantially differently, and at least some of them would manage to avoid that particular issue.

Is this lawbreaking stuff? Is it only lawbreaking if the caster willingly influenced the past? What if he accidently changes things in the past? What do you think?

(This thread is meant as a brainstorming and search for story/drama opportunities and not to shut down my player)

The Law (in the metaphysical truth sense that produces the Lawbreaker power) does not care about intent.  Screwing up because you were stupid and careless is just as bad, metaphysically speaking, as purposefully meddling with the timestream.  This was addressed quite recently (again) in the "'official' perspective on lawbreaking" stickied thread by way of a Word of Jim.
The Council, on the other hand, is slightly less consistent, but on average, even less forgiving.  If it even looks like Lawbreaking, there's a chance that you'll get a chopping.

If necessary, you could probably pull off my trick, above, using sufficiently advanced scrying spells (still functioning entirely in the present).

Accepted Compels are complications for which you are compensated (yay for alliteration).  They are essentially cost-neutral, or are supposed to be.

The closest thing we see is Denton and the other feds in Fool Moon, and even that seemed to be a special circumstance.

That wasn't anything like a 'government entity'.  That was a handful of rogue agents acting of their own accord after being manipulated, probably by the Black Council.

The closest thing to a government entity dealing with the supernatural that we see in the novels is Special Investigations, and they don't really report to anyone regarding their dealings, so it's questionable as to whether or not they would actually count as a government entity in that regard.  It could be argued that SI is, instead, a rogue department, acting with neither the knowledge nor the approval of their superiors (who at least outwardly refuse to acknowledge the existence of the supernatural, and burden SI with the cast-offs of more reputable departments).

I dislike -0 powers, but even more so, I dislike -0 powers with other powers as pre-requisites.  They seem superfluous to me; where one would otherwise be called for, I much prefer simply to use aspects.
If a character has aspects that support the use of mental evocation attacks (and they are already in possession of appropriate channeling or evocation powers), then allow them to do so using the rules you determine are appropriate.
If a character with channeling or evocation lacks aspects supporting the use of mental evocation, then they should be as unable to use them as Harry attempting, well, any of the things that Harry is simply completely unsuited to attempt.

The novels are not actually incompatible with the 'vampire barbecue incident' being a Law violation.
The effects of violations on the perpetrator are not obvious each and every time.
We simply do not have sufficient evidence to definitively say one way or another