"есть" means "there is" as well it is the infinitive form of the verb "to eat". It is part of the phrase "to have" because Russians commonly use the phrase "у меня есть" which means "near me there is". The "ест" that you find in this sentence is the conjugated form of the third-person singular, meaning "he/she eats". Or at least that's what I learned, I'm a beginner myself.
"У этой девушки грипп?" is the correct variant. "эта девушка имеет грипп?" is grammaticly correct, but sounds weird ( we seldom use the word "иметь" in Russian, and if you say "иметь кого-то" - "to have somebody" it would sound a lot like "❤❤❤❤ somebody"); "у эта девушка ест грипп?" the word "эта" should be changed accordingly to the case and "ест " means eats, so this phrase means "this girl eats the flue"
You use does in an affirmative sentence to add emphasis to it.
Note the two possible translations:
Yes, this girl does eat mushrooms - When you write a sentence like this you are emphasizing the fact that she eats mushrooms. It could be a reply to an affirmation. Take this conversation as an example: person 1 - The girl does not eat mushrooms at all!
person 2 - Yes, she does!!! I'm telling you, the girl does eat mushrooms!!
Yes, this girl eats mushrooms (note the conjugation) - This sentence is used as a simple reply to a question "Does this girl eat mushrooms?"; "Yes, this girl eats mushrooms"
Hope that helps