Overall, I think you've done a good job with this essay, but I notice that you're planning to submit it under "additional information." To the best of my knowledge, the "additional information" section is generally used to address issues that do not fit in any other part of the application- for example, adverse circumstances that might explain a sudden drop in grades (a death in the family, unmedicated depression, and so on). From what I've heard- and I may be wrong- people do not usually advise submitting an additional essay under the "additional information" section, since the Common Application already provides room for an essay. However, I suppose that the choice is up to you.
"Two years ago, when I, for the first time, went abroad, I was excited and nervous. Questions burned in my mind, but I lacked answers."
I would put this in the same paragraph as your first sentence, which doesn't really warrant its own paragraph anyway, and say "It was my first time abroad, and I was excited and nervous. Questions burned through my mind." There are simply too many commas in your the first sentence ("Two years ago [...] nervous"), and the part about lacking answers is implicit in your statement.
"I had not prepared for this."
This sentence would sound more natural as "I was not prepared for this."
"However, I frowned to this idea."
"Frowned to" should be "frowned at."
"Since I was an only daughter at home, the center of the whole family, parents seldom required me to cook myself."
"Parents" should be "my parents", since you're talking about your situation specifically, not about all only daughters and their parents.
"I determined to impress them with this delicious Chinese dish."
"Determined" should be "was determined." Also, "them" is ambiguous- are you referring to your host family?
"My friends "thumbed up" after tasting my soup and even urged me to teach them the secrets to it."
I know that you've put the colloquialism in quotation marks already, but I have actually never seen the expression "thumbed up" used before. Perhaps you should say "My friends gave me a thumbs up after tasting my soup", or something similar.