This was very interesting to read. ^^ Well done~

For the first sentence, I think it would sound more natural to say:
I live in a very quiet neighborhood called Hwawon which means "flower land."

For the second sentence, the sentence structure depends what you are trying to say.

If Hwawon is a part of Deagu, you might say, "It's located in the west region // part // side of Deagu." You can also use the word "western" in this case: "It's located in the western region // part // side of Deagu."

If Hwawon is not a part of Deagu, you can say, "It's located west of Deagu."

Basically, if you use the cardinal directions [north, south, east, west] you need to use a particle if you use a preposition before it. This is most common if you are referring to the region of a larger place, like the southwest region of the United States, or the eastern part of Incheon:
.. in the west
.. in the east
.. in the south
.. in the north
.. in the southeast

However, you do not need to use a preposition at all if you are using the word in this way:
.. west of Deagu
.. south of the post office
.. north of the football field
.. east of Manchester

For those examples, particles are required before nouns [the library, the restaurant, the soccer field] but not before proper nouns [Manchester, Deagu, Seoul, Portland, Rhode Island].

The next sentence is actually two shorter sentences separated by a comma. To fix this issue, I would make it into two separate sentences. Also, the phrase, "apartments had built" is incorrect. "Apartments had built" implies that the apartments built something. I think what you wanted to say is that apartments were built or had been built.

"It was about 10 years ago when apartments were built here," or, "It was about 10 years ago when apartments had been built here," are more grammatically correct.

Are you trying to say that apartment buildings started being built about ten years ago? If so, you can say, "It was about 10 years ago when apartments were first built here," or, "It was about 10 years ago when apartments started being built here."

In the next sentence, you need to put the article "a" before "few schools" and before "traditional market." It sounds more natural.

"It's still quiet and fairly small with only a few apartment complexes, a few schools, one small library, and a traditional market."

The next sentence is correct minus one detail; "in the behind of my apartment complex" sounds a bit clumsy. It is better to say, "Behind my apartment complex, there are a park and hiking courses that lead to the top of the small mountain." You could also say, "In the back of my apartment complex, there are a park and hiking courses that lead to the top of the small mountain."

I am feeling sick, so I have to go now. ><;; Here is a brief correction of the rest:

ORIGINAL: People who live in this area are used to hike on weekend, including me.
CORRECT: People who live in this area are used to hiking on weekends, including me.
CORRECT: People who live in this area are used to hiking on the weekend, including me.

If it's a hobby, you can say:
.. People who live in this area enjoy hiking on weekends, including me.
.. People who live in this area enjoy hiking on the weekend, including me.

This sentence would be better phrased if "including me" was located near the phrase referring to the people, and not the phrase about hiking on the weekends:
.. People who live in this area, including me, are used to hiking on weekends.
.. People who live in this area, including me, are used to hiking on the weekend.
.. People who live in this area, including me, enjoy hiking on weekends.
.. People who live in this area, including me, enjoy hiking on the weekend.

ORIGINAL: Even though it's small, multiple bus routes serve the area and subway station will be constructed in next year, so most of us who live here feel convenient and look forward to new subway station.
CORRECT: Even though it's small, multiple bus routes serve the area and a subway station will be constructed next year // within the next year, so most of the people who live here // Hwawon's residents feel convenient and look forward to the new subway station.

I will make a quick note about the use of "convenient."

Generally speaking, people cannot feel convenient. It is an adjective used to describe something inanimate. People can feel happy or fortunate or content or well provided for, as some examples. I think well provided for is the most natural phrasing in this case, but it is up to you.

Very well done!