Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:
Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.
"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.
This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.
The primary source on Favored Enemy is either Player’s Handbook in the ranger class description, or Rules Compendium if you buy its assertion of primacy. The descriptions in the Invisibility description in Dungeon Master’s Guide, Improved Manyshot in Epic Level Handbook, or even darkness, despite also being in the Player’s Handbook, are definitely not the primary source on Favored Enemy.
Further caveats, limitations, and addenda not mentioned in the primary source description are contradictions with that description. If Favored Enemy had defined itself as precision damage, the primary source on precision damage would apply, but it didn’t, which means that neither the precision damage description nor anywhere else can define it as such.
I would be inclined, in general, to follow contradictory rules as far as they go. Darkness and Invisibility cannot define Favored Enemy as precision damage in general, but it can say Favored Enemy doesn’t work in those conditions. Here, specific-trumps-general: rather than trying to redefine what Favored Enemy is (in which primacy asserts itself), they can define a special case which acceptably contradicts the general rules. So my reading of the rules as written would be that Favored Enemy is not precision damage, but it does fail to work in cases of darkness or Invisibility, and does apply only once to Improved Manyshot.
Ultimately, however, I would mostly ignore any and all rules as written that apply any more needless limitations on Favored Enemy. Of all the iconic core class features,1 Favored Enemy is one of the weakest, and that’s even assuming that it “just works” on any and all attacks against the designated foes (including, therefore, Improved Manyshot).
- Slow fall and wild empathy, if counted as “iconic class features,” are definitely weaker than favored enemy, and smite evil and trapfinding give it a run for its money too. That’s still a small list compared to all of the options.
Yes. Because skill checks are ability checks.
See pp. 173-174 of the PHB, specifically the delineation of the "three main rolls of the game--the ability check, the saving throw, and the attack roll" and the following description of skills:
Any History check is an Intelligence check: it's an Intelligence check in which you might be proficient, and covers a small subset of Intelligence.
An authoritative clarification on this matter came in The Ability Check, one of Jeremy Crawford's Sage Advice articles: