In the first question, you are asking about the ingredients of a salad. If you are specifically interested in the relative proportions of one vegetable over others, you would probably have to be pretty specific. You'll also want to make sure to pair “most” with “of” in these situations, and introduce it with the definite article.
Ж What vegetable does that salad have the most of?
It's pretty awkward, but could be understood.
A native speaker might instead ask:
What's the main (or primary) vegetable in that salad?
If you just want a summary of the ingredients (“I want to [know] what [...] vegetables [are] in it”), then a good construction would simply be:
What's in that?
Your second example is a little more difficult to evaluate. It is certainly not common to say it that way, but the construction isn't the only problem. “Better” is not a good parallel of “most”, and qualitative evaluations are generally going to go a little different from questions of quantity.
These two are grammatical, but awkward:
Ж What is there (the) most of in that salad?
and, as closely parallel as possible:
Ж What is there better of in the movie, writing or directing?
The latter is especially awkward.
You'd be better off saying this:
What was better, the script or the directing? (= Which did you like better, the script or the directing?)
Not sure I've really addressed what you're looking for here. The short answer is “yes, that sounds weird to me as a native speaker”. Feel free to add examples or ask follow-up questions.
In the answer above, a Ж ('Zhe') indicates an awkward construction that should be avoided.