Yes, it does. The formula for calculating tooltip damage without modifiers is:

(R + (D*10.6))*X

Where:
R = applicable resource
D = applicable damage (spell or weapon)
X = coefficient of the skill (unique number for every skill).

So in the case of something like Snipe, the damage tooltip (again, without taking into account modifiers like the champ system) is the result of your weapon damage times 10.6, then the result is added to your stamina, then that result is multiples by Snipes coefficient. And yes, use the stamina you read in your tooltip (so yes gear that adds stamina or facials that increase stamina affect this).

I don't get it. I just don't get it. WHY would ZOS write the tooltip like this if it actually increases damage as well:

"Affects the amount of Stamina you have available to use Physical Abilities"

Don't they think the small little detail that it also increases damage isn't worth adding to a tooltip? We're talking about adding a line of text here for crying out loud. It could be fixed in less than a minute and still they let hundreds of thousands of people play this game for a year now and many don't have a clue about what does what. Literally nobody would know simply by playing the game.

I mean, how hard can it be? *facepalm* I'm literally speechless. Do they really care this little that players have no freaking idea what they are doing without digging up information on forums? I mean, come on!