Lang-8 Is Wonderful

Today I’m going to share with you a wonderful experience I had recently.

"These days" would only be used for an experience that lasted several days, or one that was repeated several times recently.

I thought we might have misunderstood each other, or something. [Alternative: "... misunderstood each other somehow."]

"Or something" can be omitted. "Something" between "misunderstood" and "each other" doesn't make sense; what are you thinking in Japanese? The alternative I gave has a different meaning, but I'm not sure what meaning you intended.

He wasn’t learning my native language, Japanese, or even English (which I can write in better than I can write in Korean), so I had to exchange messages with him in my poor Korean.

Use "or" for a negative verb phrase:
He was learning Japanese and English.
He wasn't learning (either) Japanese or English.
The appositive phrase "which I can write in better than I can write in Korean" can be surrounded either by parentheses or by commas. Since there are already several commas in the sentence, I think it is easier to read with parentheses.

At last, I had no choice but to ask the writer/author of the book (fortunately, he was Japanese), so I found out the writer’s mail address via the Internet and asked him about it.

It was a little scary for me to ask the writer directly, because I was a totally stranger to him.

"Totally" is an adverb; you need the corresponding adjective "total" to modify the noun "stranger".

It was a grammatical mistake that even some Korean people tend to make.

"Tended to make" is possible, but it is more natural to use the present tense for something which happens repeatedly, and can be expected to continue to happen in the future.

He was indeed a nice person, since he sent the reply quickly and gave me the explanations of the grammar. [Alternatives: It was nice of him to send me a quick reply and to explain the grammar (to me).]

Even though it has a mistake, I’ll keep on using it.

I am grateful to the Korean who corrected the mistake in the first place.

The direct object of "appreciate" is usually something a person did, not the person.

About three days ago I got a correction on my Korean entry from a Korean native speaker, even though I had copied the sentence exactly from a Korean textbook.

You could also say "copied the sentence verbatim".

I thought we might have somehow misunderstood each other.

In the end, I had no choice but to ask the writer of the book (fortunately, he was Japanese), so I found out the writer’s mail address via the internet and asked him about it.

The phrase "after all" would work if you set it up better for contrast, e.g., "I thought maybe we'd be able to avoid my rather limited Korean by discussing it in English, but he didn't know English, so in the end we had no choice but to discuss it in Korean after all." Note, however, that no matter how you set it up, "after all" does not mean the same thing as "in the end". "After all" means "contrary to the aforementioned expectations". (It also has another meaning, in certain other contexts, but that's irrelevant here.)

It was a little scary for me to ask the writer directly, because I was a total stranger to him.

"Totally" (like most English words that end in -ly) is an adverb, so it would have to modify a verb, an adjective, or another adverb. For example, you could say "I was totally a stranger to him", with "totally" modifying the verb "was". Adverbs that modify the main verb are fairly flexible about where they can go in a sentence, but you can't inject them between a noun and the article that modifies it. The only thing you can put there is an adjective that modifies the noun (or sometimes another noun that goes with the noun to form a noun phrase). The adjectival form of "totally" is "total". This is a general pattern: adverbs that end in -ly are usually formed by adding the -ly to an adjective, e.g., "slowly" is the adverbial form of "slow", "happily" is the adverbial form of "happy", and "facetiously" is the adverbial form of "facetious". To get back the adjective, simply remove the -ly ending.

It was a grammatical mistake that even some Korean people tend to make.

"Tented" is a word, but it's related to the noun "tent":
https://www.google.com/search?hl=en&tbm=isch&q=tented

He was indeed a nice person, to send such a quick reply and explain the grammar to me.

Today I’m going to share with you a wonderful experience I had the other day.

About three days ago, I got a correction on my Korean entry from a native Korean speaker, even though I had copied the exact sentence from a Korean textbook.

I thought we might have misunderstood something each other.

He wasn’t learning my native language, Japanese, or even English (which I can write in better than Korean), so I had to exchange messages with him in my poor Korean.

After all of that, I had no choice but to ask the author of the book (fortunately, he was Japanese), so I found out the writer’s mail address via the Internet and asked him about it.

It was a little scary for me to ask the writer directly, because we are total strangers.

It was a grammatical mistake that even some Korean people tend to make.

He said that he is going to correct the mistake when the next edition comes out.

He was indeed a very nice person, since he sent me a quick reply and gave me the explanations of the grammar.

Also, "...sent me a quick reply and explained the grammar to me."

The book is the only Korean textbook I have now, and it’s my favorite.

Even though it contains a mistake, I’ll keep on using it.