Is an Initiative check considered an ability check for the purposes of things like whether or not it gets the bonus from a stone of good luck?
An initiative check is a Dexterity check.
Under some definitions, all d20 rolls that include an ability modifier are ability checks. It's probably best to assume that an ability check is a d20 roll that includes only an ability modifier.
people seem to rule that it's an ability check but not an 'opposed' ability check.
The game really doesn't have many ability checks built into it. The GM has to go out of his or her way to add them to the game. I researched this at one point during 2014. Here's everything I found. It doesn't seem like a lot. The reason, supported I beleive by the sporatic array of checks found below, is that the designers never really devoted energy to making it a cohesive part of the game. There is GM fiat, and there is this list. I may have missed a few.
-Strength: breaking or bursting items, resisting great winds, breaking certain forms of entanglement
-Dexterity: initiative, untrained slight of hand
-Constitution: heal when disabled, stabilize when dying, holding breath, forced movement, chases, starvation, thirst, suffocation, drowning
-Intelligence: the contact other plane and maze spells
-Wisdom: animal using hunt trick if untrained in survival, detect poison spell, sailing checks
-Charisma: implanting ioun stones, dealing with called creatures, certain enchantment spells, controlling a berserk flesh golem
Under some definitions, all d20 rolls that include an ability modifier are ability checks. It's probably best to assume that an ability check is a d20 roll that includes only an ability modifier.
I don't think this is right at all.
If, for example, you have the Muscle of the Society trait, and you try to break down a door, is the check to break down the door no longer a Strength check because the trait is giving you a bonus?
The basic mechanic itself uses only a d20 roll and the ability mod. Some characters have a bonus to the check.
The basic mechanic itself uses only a d20 roll and the ability mod. Some characters have a bonus to the check.
I'm not sure if you're referring to initiative checks here, but this post is true of initiative checks.
Is an Initiative check considered an ability check for the purposes of things like whether or not it gets the bonus from a stone of good luck?
Yes. This is pretty clear from the text of Initiative (it flat out says it's a Dex check), but if you need more proof:
At 20th level, an empiricist's powers of reason and deduction become almost superhuman, and he is able to use them in nearly all aspects of life. At 20th level, an empiricist can use inspiration on all skills (even ones he is not trained in) and all ability checks (including initiative checks) without spending inspiration.
This ability replaces true inspiration.