Second attempt (original below):
Ears
I have more to do with cooking than you might think.
Ears of corn? Also cooking does make noise, especially frying or sauteing
Now let us discuss the ocean, try not to sink.
You can hear the ocean in your ears
Then between quizzes and sandwiches I like to rest.
your brain (which handles quizzes) is sandwiched between your ears
The beginning of eating, a feast is the best.
Half the area is needed, we aren't quite done.
We wrap up wondering if I am more than one.
These three lines are the wordplay. "e" is the beginning of eating. "ar" is half of area, and we add an "s" to make it more than one. Still not sure about the "feast" part
Now as you guess, know that I am quite small,
Doesn't really matter if you are short or tall.
Like my previous rationale, ears are small no matter what your height (though some people do have smaller/larger ears)
Two types of clues to guess one type of me.
What on the earth could I possibly be?
--- ORIGINAL GUESS ---
Here's an answer:
eyes
I have more to do with cooking than you might think.
If food looks good it can have an effect on our appetite and desire to eat it
Now let us discuss the ocean, try not to sink.
Haven't gotten this line yet
Then between quizzes and sandwiches I like to rest.
Your eyes are between your brain (quizzes) and sandwiches (mouth)
The beginning of eating, a feast is the best.
This gives you an "e", also your eyes can "feast" on a view
Half the area is needed, we aren't quite done.
Area is x*y, so "y" is half of it (this part's shaky)
We wrap up wondering if I am more than one.
"eyes" is a homonym for "I's" - are "I's" more than one person?
Now as you guess, know that I am quite small,
Doesn't really matter if you are short or tall.
Eyes are a small part of the body, no matter your height
Two types of clues to guess one type of me.
What on the earth could I possibly be?
We have the wordplay ones and the literal clues