저는 이야기를 쓰는 거걸 너무 좋아해서 여기에 왔어요.

Don't omit particles unless you're absolutely sure you should be omitting them. In this case, it's more natural not to omit 를(-ㄹ).

이 웹사이트는 처음으로 사용해서하는데 잘 부탁드려요.

해서 isn't necessary wrong, but I think 하는데 would be more appropriate as the clause in front (이 웹사이트는 처음으로 사용한다) is more like background information rather than a direct reason.

이 이야기는 제 상상으로 했썼어요.

오랜만에 안 써쓰는 거라서 실수가 있을지 몰라요.

오랜만에 하다 means you're doing something for the first time in a while. 오랜만에 안 하다 means you've consistently been doing that thing, and it's your first time in a while NOT doing it. So, saying "오랜만에 안 쓰는~" here does not work.

If you wanted to say 안 쓰다, you could rephrase this as "오랫동안 안 써서" or, better, "안 쓴 지 오래돼서" (= "쓴 지 오래돼서").

Also, I would personally say 있을지'도' 몰라요, but that is minor, so I did not include it as part of my correction.

고칠쳐 주실 수 있어요?

If you just say 고칠 수 있나요, it sounds like you're asking whether or not we'd be capable (rather than whether or not we'd be willing to). Also, don't forget to add -시- for politeness.

암튼 시작해요?

This isn't wrong, but maybe consider saying 시작할게요 instead.

한 젊은 남자는 오랫동안 혼자 살았다.

Again, don't omit particles too much.

친구가 없긴 하했지만 상관없다고 생각했어요다.

Tense mismatch (살았다, 하지만)
You absolutely cannot (in any case) mix the plain form (formal non-polite, ~다) with the 어요 form (casual polite). Make sure to stay consistent.

많이 일이랑일도 많고 취미가도 있어서 친구를 만들기 위해 시간이도 없고 필요없어도 없다고 생각했다.

Is this a monologue? If so, then I guess -어 is fine. Otherwise, though, you again cannot just mix manners of speech.
많이 is an adverb, so 많이 일 is not grammatical. 일이 많이 있고 취미가 있어서 would be grammatical, but 많고 would be better in that case.
Also, given the parallel structure, the -도 grammar point would be appropriate.
친구를 만들기 위해 is similarly an adverbial, so 친구를 만들기 위해 시간 is not grammatical. It should be 친구를 만들기 위한 시간, but then you can just say 친구를 만들 시간.

새로운 하루, 새로운 기회."

I think you kind of need the comma there for this to work. Even then, though, it feels quite funny and English-y.
I think something like "오늘도 힘내자!" would suffice if you want something that actually sounds like spoken Korean.

남자는 하루하루 일하면서 노래를 부르는 거걸 좋아해했다.

Particle omission
Again, this seems out of place. Was it meant to be a monologue or an internal thought?

그 다음은, 그가는 음식을 준비하고 점심 먹으러 나무 밑에 앉았고 아서 점심을 먹었다.

먹었다 pretty much always needs an explicit object (except maybe when you're answering a question and the object was given in the question), so that word order is better for the latter part of the sentence.
The transition is kind of awkward. 그 다음 is slightly better than 다음에, but it's still pretty bad. If you could clarify what the previous sentence means, maybe I could come up with something better.

근데 오늘은 이상한 소리를 들었다.

This is correct, but I think Koreans would be more likely to use passive voice here and say 이상한 소리가 들렸다.

남자는 바로 일어나서 음식을 떨어졌는데 그 소리로트리고 소리를 향해서 달렸다.

Unclear subject; subject of the previous sentence was 어떤 모르는 사람, not the man.
음식을 떨어지다 is not grammatical; 떨어지다 is an active intransitive verb (물건이 떨어지다). 떨어트리다 is a causative transitive verb (물건을 떨어트리다).
I don't think -로 is enough to say "run towards the noise."

나 아는 사람이 없는데.

This is fine because it's spoken but I would advise you keep the 는 after 나 as a learner.

빨리 가야서 내 집 시키야을 지켜야 돼."

Not entirely sure what this was supposed to mean. Was 시키야돼 a typo?

요새 어때요?

This is very Englishy. 다들 잘 지내시나요 maybe?

항상건 건강하시고 즐거운 시간 보내행복하세요 ~

즐거운 시간 보내세요 = have a good time; "always have a good time" sounds strange.
행복하세요 is not strictly grammatical (because 행복하다 is an adjective), but that is what we would say.