Are these sentences correct? (Cover letter)

I've been studying at Fake Lesson Center, a school that ensures students get proper training for the ENS.

"My two years of associating university and preparatory classes have helped/made? me (to?) become a more hardworking person."

My two years of attending university and preparatory classes have helped me become a more hardworking person.
Two years of attending university and preparatory classes have made me a hardworking person. (Slightly more natural)
Two years of attending university and preparatory classes have taught me the value of hard work. (Better nuance)
Two years of attending university and preparatory classes have broadened my knowledge and taught me the value of hard work.  

"This is for me the perfect opportunity of studying what I'm interested in"

This is a perfect opportunity for me to study what I'm interested in. (More natural than your two options)

Osaka City in Japan

In general titles are capitalized with exception of grammatical articles.  You can get away with capitalizing everything three characters or longer.

Food in this context is an uncountable noun, so it cannot be pluralized.  I can eat "two meals" or "too much food", but cannot "eat two foods".

Central is an adjective, therefore Shinsaibashi can be central to Osaka, or Shinsaibashi can be known as the central district of Osaka.

I recommend that you might wanna go to Universal Studios Japan. I think it’s the best amusement park in Japan. Harry Potter area is the most popular area there. Its castle is very realistic and the roller coaster and riding machine are very exciting. I like the Jurassic Park area too. The roller coaster and riding the boat are very interesting. Osaka is famous for its food, such as Takoyaki and Okonomiyaki. They are so delicious.

Saying "I recommend that you might want to..." is redundant.  Say either "I recommend that..." or "You might want to..."

I recommend "Harry Potter is the most popular area there."

I recommend "Riding the roller coaster and riding the boat are very interesting." or "It is fun to ride the roller coaster and boat."  It is awkward to mix a noun and an action the way you've done.

You can go to other famous places easily from Osaka, which are Kobe, Nara and Kyoto. There are many traditional Japanese shrines and temples in Kyoto. If you go to Nara, you can see Big Japanese Buddha which is called Daibutsu.

"Which are" implies an complete list, but Japan has more than three other famous places.  Use "Such as Kobe, Nara, and Kyoto" or "Including Kobe, Nara, and Kyoto" instead.

Big must be capitalized because it is part of the name.

I'd also recommend that you break your writing into paragraphs, as has been done in this correction.

1) Someone have prized up the door made by plywood.
Someone has pried up the door made from plywood.
Someone (had/has/-) pried up the plywood door.

3) A butterfly is fluttering above the flower beside the placid lake.
Correct

4) I asked him that what day of the week today in the harmonious morning.
This whole sentence is quite unnatural.  This is better, but still unnatural: "I asked him what day of the week it is this harmonious morning."

Harmonious is generally used to describe two or more things/people getting along.  For example "Strawberry and banana are a harmonious combination", "Vanilla is a harmonious flavor, it goes with everything", or "Today is a harmonious day, nobody is fighting".

5) When I kept watch, I dozed off.
Correct but very ambiguous.  This could mean "Every time I kept watch, I fell asleep" or even "The instant I tried keeping watch, I fell asleep".  "I dozed off while keeping watch" seems more natural.

6) I fly into a rage as if I blow a fuse.
Your sentence means "I get angry as if I get angry" since both sides have the same meaning.  

7) She brings over a shirt with a stain of coffee.
This is unnatural because it could also mean "She brings over a shirt and a coffee stain."
She brings over a shirt stained with coffee. / She brings over a coffee-stained shirt.

8) She prodded my sore spot intently.
Correct

9) I rummaged in a chest of drawers to come up with a document.
"I rummaged through (the/a) chest of drawers to come up with a document" is more natural.

10) She scoops a coffee and hot water into a cup and takes a couple of puffs.
This is a weird sentence because you can scoop coffee powder, but must pour water.  Puff is an unexpected word in this context because "taking a puff" generally describes sucking in and blowing out cigarette smoke.

She scoops coffee into the hot water, and takes a couple of sips.
She pours hot water into the cup, adds a scoop of coffee, then cools it with a (few/couple of) puffs (of air).
She adds coffee and hot water to a cup, then blows a few puffs of air to cool it.

You are right about "How are you?" being superficial.  The expected answer is often just "Good, thanks", so neither party takes the greeting literally.  Once in a while a complete stranger will actually go into their life story, which is awkward.

I tend to prefer "Nice to meet you", "Good/Nice to see you again", or "It's been a while" when greeting strangers or acquaintances.

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These two sentences are so closely related that I think a comma is necessary.

I think It is so awkward to say "Hi! How are you?" - and giving a hug or kiss on the cheek is just as bad. Well, usually you don't give real hugs and kisses, right? It's actually just a slight hug and touch on the cheeks. Of course I'm not saying I'd like to hug and kiss people for real, that would be way worse.

You've already said that you are writing about yourself, so including "I think" or "I thought" too many times is redundant.  This first one is marginal, and works better if you say "I think it is too awkward..." or "I find it (too/so) awkward...".

Also, I used to avoid saying "how are you", because I thought it was so superficial. I didn't want to know how the person was doing, so it would be just from mouth out if I asked them. Nowadays I ask "how are you", just because it's considered more regardful, but I still feel a bit bad about it. I wish I didn't have to greet people.

You've moved on to a new idea so I think a paragraph is appropriate.  In casual writing 1-5 sentences per paragraph is ideal.
The yellow sections are correct and understandable, but not natural expressions.  I'm having a hard time coming up with good English equivalents, I might have said "shallow" and "friendly" respectively, but those are kind of dry.

Maybe "I didn't want to know how the person was doing, so the question had no heart."

1) Correct

2) Lights that impair night vision are prohibited on the adjacent highway.

"At" refers to a single point.  A highway, path, or road does not have a center, so "on" or "near" are more natural.

3) Migratory birds rest above the moonlight shimmer of the ocean. (Some birds can sleep while flying.)

The original sentence is gramatical but does not make sense.

4) A number of birds collide with broadcast towers.

5) Due to the ordinance, events with a high likelihood of accidents are forbidden.

6) Coincidentally, he looked skyward.

Skyward is already a direction.

7) After the retrofit, we will block all trespassing into this area. / The retrofit will block all tresspassing.

"Any kinds" should be "all kinds" or "any kind".  The original sentence also seems to have a tense problem.

8) There is an entirely different set of stars in the north and south hemispheres. / The northern and southern hemispheres contain an altogether different set of stars.

Altogether is a hard word to use naturally.

9) Correct, but "Two out of five" or "two in five" are more natural.

10) At this rate, it's going to be too much stimulation for you. / At this rate it will be too much stimulation for you.

11) Thus, mere reinforcement is less rewarding.

I do not fully understand the original sentence.

12) Correct