Store Greeting - Pre-Purchase: "Earthmother Guide You."
Purchase: "It is my turn now."
Store Greeting: "Ish-ne-alo por-ah, may the Earthmother smile upon you."
Ready: "For honor!"
Intro - Greeting Thrall: "Thrall, my old friend...I am ready to fight by your side once more."
Intro - Greeting E.T.C.: "Kids these days and their rock music..."
Intro - Greeting Ana: "You may be human, but your care for the young rivals that of my own."
Intro - Greeting Murky: "Hail, little one. You're quite far from home, aren't you?"
Intro - Greeting Nazeebo: "I can see the ancestors and you have much in common, Witch Doctor."
Intro - Greeting Garrosh: "I will fight beside you, Hellscream, then you and I will settle our differences afterwards."
Intro - Greeting Sylvanas: "Don't make my people regret this choice, Windrunner."
Intro - Greeting Sonya: "Ah, this one here has the heart of the wilds, beating proud and free"
Intro - Greeting Dehaka: "I will be keeping a close eye on you..."
Intro - Greeting Positive: "We will make our stand here."
Intro - Greeting Negative: "It is clear you will need my help..."
Intro - Response Positive: "I will fight with honor and pride by your side."
Intro - Response Negative: "Fate will be your judge."
What 1: "Mhmm?"
What 2: "Yes?"
Humorous 1: "There are so many interesting people here in the Nexus...quite a few of them keep referring to me as 'The Cow King', whatever that is."
Humorous 2: "I'm hoofin' it as quick as I can, hahaha..."
Humorous 3: "I would very strongly recommend not trying to grab the bull by the horns, here."
Humorous 4: "Our kin are all over Azeroth, and we all have many different cultures and appearances...but there is one thing we can all agree on; our hatred for Harpies!"
Move 1: "Let's go."
Move 2: "Move aside."
Move 3: "On it."
Hearthstone: "I'm returning to the fountain."
Getting Healed: "Ancestor's watch over you."
Attack 1: "Attttacckk!
Attack 2: "*roar*
Attack 3: "Strike them down!"
Killing a Hero 1: "Learn your place, whelp."
Killing a Hero 2: "This old bull still has a few tricks."
Killing Cho'Gall: "You're not Stonemaul, that's for sure..."
Killing Falstad: "Struck down like a harpy."
Kiling Garrosh: "This is how Mak'gora would've gone..."
Killing Malthael: "The souls of my ancestors are safe once more."
Killing Thrall: "Get back up, Thrall!"
Death: "Ancestor's guide me."
Revived: "I am not quite dead yet."
Ping - Kill Hero: "Strike this hero down."
Ping - Capture Mercenaries: "These mercenaries shall join us."
Ping - Help Ally: "Our ally needs help."
Ping - Help Me: "I won't last much longer..."
Ping - Destroy Fort: "Crush this fort under hoof."
Ping - Defend Fort: "Protect this fort from destruction."
Trait: I've always felt that Cairne would make the most sense with a trait like this that rewards him for playing close to his team, and I think what you have here makes enough sense. It's a fine trait, but the way it would be written should be more like this to match HotS standards:
"Cairne recovers 1% of his total health per second for each nearby allied hero, up to a maximum of 4%."
Though for shorthand, you might want to consider something like this:
"For each nearby allied hero, Cairne receives a stack of Earthmother's Grace, up to a maximum of 4. For each stack, Cairne restores 1% of his maximum health per second."
This would just make it easier when writing talents, as you can just write
"For each stack of Earthmother's Grace, blah blah blahX"
which is just easier to parse.
I also imagine there'd be a little something in the lower left corner that lights up for each nearby hero.
Q: This one's a fairly reasonable tool, but I think that the slow amount is just too trivial for a tank. 10% slow is fairly meaningless, and will do nothing to save your team or win fights. My personal suggestion would maybe be to give it a longer-lasting slow and a shorter cooldown, wherein Cairne is able to repeatedly apply the effect, stacking it over and over. It may have to decay in order to make the duration not be insufferable, but it would synergize well with his trait, as the longer the fight goes on, the more things tip in Cairne's favor. You could maybe have the slow be applied in the form of marks, but either way, you open the door for some talents that reward maximizing your slow on a particular target.
W: I like stomp. My own WIP for Cairne has this move, but as a Root instead of a Stun, and targeting in a specific direction. A point-blank bit of hard-CC with a low cooldown makes a lot of sense on a hero of this ilk. "Don't come near me or my friends or I will bury you" is a fitting tool. It's somewhat more defensive by nature, but can be used to secure kills, similar to Uther's Divine Storm. That said, while everything else in your moveset errs on the side of being too weak, this move errs on the side of being too strong.
E: This one... unfortunately feels out of place. It reminds me of Garrosh's Gorehowl talent where he gets resets for killing minions, which is a fun tool and all, but given you have Shockwave to fill the niche of 'low cooldown damage tool' this one feels redundant, and may keep Cairne from really fulfilling his job as a tank, as even with the Q changes, he just doesn't have the tools he needs here. If I had to suggest anything, it would be to bump this move's power up and turn it into a heroic, or maybe make it a lategame talent for Stomp (for example: "Stomps deals bonus damage to foes below 50% health. If stomp kills a hero, its cooldown is reset" could lead to some amusing scenarios where Cairne just repeatedly stomps his hoof on an entire team as they die one after another). Personally I'd suggest maybe instead giving Cairne a mobility tool, maybe paired with AoE disruption.
R1: I get that you're trying to go for a hero with two passive heroics, but this one is just... not really much of anything. The concept just doesn't have enough to it, so you could easily turn it into 7 or 13 talent.
R2: The only thing I'm going to say about this one is that I think you could very well add more substance to this. I see you're trying to balance the fact that most resurrect mechanics are 20s by giving it a long cooldown, but I think 5 minutes boils down to only getting to use this effect twice per match at most. Once again it could be a 13 talent instead, or maybe you could turn it into a quest heroic of sorts, or having built-in cooldown reduction mechanics tied to PvP, that way good players can res more often.
I do like certain aspects of this kit enough that I actually want to implement them into my own Cairne moveset, but I think they just need a little bit more of a push so that he can have an impactful tank gameplan.
A lot of your suggestions are exactly what I'm looking for, because it makes me think about things differently without entirely replacing the core concepts of what I envisioned as his design. Based off your suggestions I know right away what changes I want to make on the his first three abilities. His ults are what get a little tricky for me though, because I imagined the idea of having passive ults being pretty cool, a sort of 'power without having to worry about actively using it' scenario. But looking at the two of them in a new light perhaps they're better serving other roles (i.e. talents). It's highly likely I'll come up with two new ults for him and use his current ults as talents (with a bit of adjustments here and there).