Maybe.
?
There is nothing in those rules that say that having a Climb speed foregoes the need to make a check to successfully climb a surface. - Just that it doesn't cost extra movement.
If a druid turns into a spider and wants to walk along the wall, but the DM doesn't want them to, the DM can just set a DC that the spider can't make.
- This is pretty much as deadly as the current rules. Unless you die from massive damage, you are down to making death saves, which generally mean that you have a 50/50 chance of dying in a few rounds.
As far as I'm aware, 5e has always used Death Saving throws.
Might work for the Barbarian. Not so good for the Rogue.
Not to mention that you probably want to keep it friendly to you, so Handle Animal checks might be better than a subset of attacking it.
"The DM has final say on whether your assistance is possible." - Its literally in the rules that you're talking about.
The check in question is the one Bob or Dave is making. It looks like Alice needs to use a skill or tool she is proficient in, but making an extra roll to determine whether Bob gets an extra roll isn't very streamlined.
Nope. You have to expend your Heroic Inspiration to use it, and you only choose to use it immediately after rolling the D20, whether you know if that roll was high enough or not.
If the monster dropped you to 0 HP, you are not only Incapacitated, you are Unconscious. Your speed is 0 and cannot change, you are prone, and you automatically fail Str and Dex saves.
Both situations where, according to these rules, they couldn't knock out their opponent.
When you're about to take someone down, punching them out with your pommel rather than stabbing them is a pretty classic move. Unless the DM is unusually precious about narrating death scenes for their NPCs or insists that every attack roll represents a single discrete attack that must be thoroughly described before rolling, I'm not seeing an issue.
Any sort of combat interrupts the rest, as well as a number of other things including spellcasting.
If the area that the party are trying to rest in isn't safe, they are unlikely to get to finish that Log Rest anytime soon.
Would you prefer to insist that the party have to perform some specific actions to rest rather than just setting limits to how strenuous it can be?
. . .
This is really starting to sound like a table issue rather than a rules issue.