I very much doubt it, but does anyone have a number of particular terms used by elves or goblins (preferably elves, or really any other speaking races) throughout the storylines? The "shaya" and "Ashaya" thing is all I really recall in recent memory.

You mean... words?

There aren't any.
Shaya and Ashaya weren't even elvish, I don't believe.

Then what was Shaya/Ashaya?

It was a weird Joraga thing.

So that's the extent of elvish lingual shenanigans?

Was it? Because it was a human who named her that.

Feel like that sort of thing is different from plane to plane, but none really come up for MTG. I remember in some story, the elves call humans something that translates to "short lives" or something like that.

Plucking the chicken
—Elvish expression meaning "flinging the monkey"

Flinging the monkey
—Giant expression meaning "throwing the spear"

Throwing the spear
-Vedalken expression meaning "salting the wound"

Are there any Zendikari terms that seem pretty much catch-all? I'm really bad when it comes to naming, and trying for a Zendikari character. So going for something that could be incorporated into a name, similar to the concept of a greedy Ravnican naming their child "Zinobah" or something to that effect.

Better question, what lingual culture shines out the most on Zendikar?

That's a stilted question dependent solely on where you are.

What do you mean?

Whole the cultures intermingle a little, there's not really anything like a truly cosmopolitan area where everyone interacts enough to get a cross section. Sea gate is the largest city on the world, but it's still really essentially a merfolk city. You'll find humans there, but not goblins or elves in any numbers worth counting.

Kor are completely nomadic so you won't find them contributing to any one place and the elves are purely elvish anywhere they are settled. Etc etc.

Oh okay, that was something I was hoping too, but I meant does Zendikar take anything for its naming of people/places in a similar way to Ravnica, Innistrad, or Theros?

We can't really tell anything about the naming convention from the locations we see.