Bonuses "Cards' v.s. "Villian' ????

If it specifies the villain character card, or something like that, then it only applies to cards that say "Akash'buhta deals...". If it says "villain targets" then it affects any card with health, and so includes the limbs. 

Mountainous Carapace only applies to Akash'bhuta and not her limbs, but needs to be taken into account when she deals herself damage when her limbs are destroyed.

Living Rockslide and Arboreal Phalanges both deal their own damages, meaning any +1 damage Akash'Bhuta does is not applied to her limbs. The DR the Mountainous Carapace gives to Akash'Bhuta also only applies to Akash'Bhuta, no limbs. So yes, there is a difference, it actually is like this for any villain that has minions. Citizen Dawn's minions all deal their own damage, but Blind Blast has herself do it.

However, I am not sure what environment you are playing in, as I can't think of anything that just increase the villain's damage. Perhaps Freedom Tower with Entry Point, which increases Villain Target damage. Which is any target from the villain deck, so the Rockslides and Phalanges would get a +1 for that.

There is indeed a difference. With "Living Rockslide", it is the Living Rockslide that is dealing the damage. The wording of the environment card matters; many of them say that "villain targets" deal +1 damage. Both Akash'bhuta and the Living Rockslide are villain targets, so they would both get +1 damage. Sometimes you'll see a card that specifies a bonus to the "villain target with the highest HP"; that will be Akash'bhuta (and not the Living Rockslide) until the very end of the game.

Were you playing A'B on advanced? If so, on one side she has an ability that reads "Reduce damage dealt to villain targets by 1". That would include both Akash'bhuta and the Living Rockslide.

A "target" is anything that has HP.

EDIT: ninja'd, twice!

Both Entry Point in Freedom Tower, and Villanous Weaponry in Wagner Mars base increase damage dealt by Villain Targets. I can't find any that increase damage dealt just by the villain.

Edit: Legacy's Landing Pad increases damage dealt by the target with the highest HP, which for much of a game against Akash'bhuta would be her.

(Really enjoying the card spoilers spreadsheets!)

Tendrils of Madness could also do it.

It's actually quite useful if you can buff Akash'Bhuta's damage but not that of her limbs, so that she takes more damage when you break them but they don't rdeal extra damage when they go off (the two that deal damage, anyway). By Any Means is a good one to put on her, for example.

The source of damage on a card is always described, I believe - if it says "this card" then it only applies to that particular card. If it specifies a named character (like the villain/hero whose deck the card belongs to), then it's that target which deals the damage (or gets whatever effect is being applied by the card in question). About the only possible exception I can think of is Horrid Cacophony, which doesn't specify who's dealing the damage - it is, however, the Matriarch herself and I think this was errata'd onto the card in the recent re-release of Rook City (in its combo pack with Infernal Relics).

Oh, I was reading it as a card that only increases the Villain Character Card's damage, not something that was currently only increases Akash'Bhuta's damage, but could change over the course of the game.