This isn't the proper way to roof, right?

The filler strips shouldn't be a problem. It's hard to exactly match old stuff, and once painted, it won't be very noticeable. Getting an exact match would likely be more expensive (custom milling) and unless you requested and paid for that, you probably got what you paid for. It would have been nice if they explained that to you though.

The gap behind the fascia where light is shining through is probably a problem. You need to look at the area above that to see how the light is managing to shine through there. Ideally, you would have also liked to have that fascia primed on all sides before installation, but that's not absolutely required. Are they finished working on the roof? If not, that could explain the light coming through.

The flashing (or lack of) at the wall is probably your biggest potential issue. There may be something there that we can't see, but I would be concerned about that. As far as the nail heads go, that's not uncommon but they should be sealed. Personally, I would have cut and glued a strip of shingles over that.

That shingle in the last pic may be normal (it's hard to tell). To get the random look, some of the shingle manufactures will actually end a shingle with a very small piece of laminated shingle like that.

It looks like you got a budget shingle job. You don't get perfection with a budget job so don't nitpick everything, but you should have expectations of a roof that's installed to code and isn't going to leak. The wall flashing is potentially a serious problem and should be checked out (if you can't see any flashing there, call the roofer out to explain). And I have no explanation as to how you can see light behind your fascia if the roof is finished, so that should also be checked out.

Pic # 1 ; A coat of paint should work.Remember the old has been subjected to the elements and has bowed and is not as flat as the new.So without replacing everything there will be a difference between the 2.I am really not liking the way they intersected the 2 connecting ends.I like to miter mine,,,much cleaner and nicer.But it looks like that's the way it was before it was repaired.

Pic # 2 ; Looks like a rookie mistake of knife handling.Chances are a shingle was cut over this one and the blade cut both shingles.Best thing to do should have been to use the sliced shingle in another area requiring a shorter shingle.Either that or the shingle was put in really tight causing a lift to that part of the shingle and instead of replacing the shingle they sliced that area causing the shingle to relax.

Pic # 3 ; It looks to me like there is flashing but instead of being exposed its nailed over with 2 courses of shingles.And its missing the top strip and that nail blow through looks fairly nasty.

I appreciate all of your responses. So much!

Those are just three examples of the MANY problems we're seeing on the house. The fascia boards are uneven and don't line up with one another at corners, shingles are sloping instead of straight in some places, they've left the job site a disaster (plastic shingle bags stuck in our trees, piles of new shingles stacked on the roof for days, and they keep dropping debris on our window AC and into our storm windows), and they aren't using the materials we were quoted in the contract (in one case, cutting straight shingles instead of installing proper GAF ridge caps).

We asked for a historically accurate repair, and we are paying SERIOUS money for this roof. Over $14k including the tree damage repair. It's a tough roofing job-- irregular roof with an extremely steep slope-- and we were cautioned about that when we had the house inspected before we bought it.

But I won't accept anything less than a stellar job for what we're paying. And this is barely even acceptable, much less stellar.

Thank you guys for giving me the confidence to be demanding with the roofers. I'm not letting them back on the roof until we have a long, long talk! (If at all.)