1) Backgrounds. They just don't contribute enough to the character's abilities and feel tacked on.
I disagree... mostly. At 1st level, I think Backgrounds make for a great addition to the character. I agree that they very quickly fall by the wayside, though. I think I'd be inclined to suggest that at 5th level they should consider having the character's Background be superseded by something else (maybe Faction at 5th level, then a Paragon Path at 11th, and Epic Destiny at 16th?).
2) Inspiration. Half-baked idea that is literally never remembered. Unless you have people always fishing for bonuses in annoying ways.
Yeah, I really dislike this as written. I've taken instead to giving each player an Inspiration token to use on any one roll, once per level. When used, it cancels Disadvantage and then grants Advantage on that roll.
Which isn't an ideal solution (far from it), but has the virtues of being simple and easy to remember.
3) Treasure Hordes. This is in the DMG, and there is actually a recommended schedule for awarding magic items and treasure. Too bad no official products ever used these guidelines.
Never used these. A shame, really - I did think the 5e DMG was pretty good, but in practice it's turned out no more memorable than the DMG for any other edition (except maybe 1st).
4) Advantage/Disadvantage. +5/-5 is too big of a modifier for most conditions. Flanking is lethal against the PCs (so we didn't use it).
I disagree. IMO, one of their best decisions in 5e was precisely that - getting rid of the host of minor modifiers, and replacing them instead with a single, big modifier. Meaning that it should only apply when it's really significant and also that when it does apply it is, indeed, really significant.
5) Bonded Accuracy. A good idea in practice, except that it turns monsters into bags of hit points.
Again, I disagree - this is a big factor in 5e being a very robust system (as far as I've been able to tell).
That said, it is important that monster design takes this into account, precisely to avoid the "big bag of hit points" effect. And too much of 5e monster design is pretty dull.