Re: #1, I've tested BT Charter Pro, and ITC Charter Com (Std), and a free version of Charter in ttf, and they don't show the smeared glyphs. So Acrobat seems to be a problem displaying the Apple version of the font in particular (which seems to be BT Charter Std packaged as .ttc). This is unfortunate, because the Apple version is the only one I can get to show up in Microsoft Word (Office 2011).
But here's #4: New Century Schoolbook LT Pro, set in 12 pt, screen grab at 125%. Acrobat on the left, Preview on the right. It's a little hard to see at smaller magnification, but it has extremely uneven color, including superdark capitals, almost like it's a bunch of different faces mixed together.
Re #2 & #3, here is a zoomed screen grab of Melior 9.75 @ 125% in Acrobat. Notice the dancing baseline. Less noticeable here, but visible at smaller magnifications, is the way the tops of the "r" and "n" also seem too high compared to x-height. Seems the way the bottoms of counters "overshoot" the baseline and the ways the tops of counters extend above x-height is rendered too strongly.
If you compare Preview's rendering of the same font and magnification, the round bottoms of the counters that overshoot the baseline are much lighter, producing a far more even and better looking baseline, seen here:
Re #1, I have now also tested the version of Charter that ships with High Sierra 10.13.4 (v. 14.0d2e1), where it is again made available to users in all apps (it had been hidden from some apps in some earlier versions of macOS), and it still shows the same bugged display in Acrobat but not Preview.
I have filed a bug with Adobe, please vote for it here: