At 6 o'clock, my aunt and I went on a walk for a short distance,but because it was raining, we only walked a little bit.

If you say "At 6 o'clock my aunt and I went on a short walk because it was raining." It seems like you went on a short walk due to the fact that it was raining.

I finished reading a book which was written in English.

The tittle was "Daisy Miller".

After I finished reading it, I checked the contents in Japanese.

It was incorrect for me to did not understand it correctly.

The real story was sad and love romance, but I thought theI had thought the story was a sad romance, but it was actually a story about a flirtatious woman.

After lunch, I faellen asleep byat 7 PM.

You can say "After lunch, I had fallen asleep by 7pm" but it does not sound very natural.

When I waoke up, I was surprised by the time.

You could also probably say "When I woke up, I was surprised at the time."

I thought I could do something nothbefore falling to asleep until next day.

I was a little confused by this sentence so my correction may be incorrect. However, are you trying to say that you had intended on doing work / studying but you fell asleep instead?

But I went to bed too soon.

Did you mean "I fell asleep too early?" 「早く寝ちゃった」?

Yesterday was a very lazily day.

I will work hard next daytomorrow!

At 6 o'clock, my aunt and I went on a walk for a short distance,short walk because it was raining.

The way you wrote it is correct, but you wouldn't hear people say that.

I finished reading a book which was written in English.

The tittle was "Daisy Miller".

After finisheding reading it, I checked the contents in Japanese.

I would personally say:
"After finishing the book, I looked at the plot in Japanese."
But the way you wrote it works too.

It was incorrect for me to understand it. understood the book incorrectly.

Saying "It was incorrect for me to understand it" means that you did something wrong (legally, morally) by understanding it.

The real story was sad and loveabout a sad romance, but I thought the story was about a flirtatious woman.

A "romance" is a love story, so you don't need to say the word love.

After lunch, I had fallen asleep by 7 PM.

You can also say:
After lunch, I fell asleep at 7 PM.

When I wake up, I was surprised by the time.

I thought I cwould do something nothingn't be able to sleep until the next day.,

Bbut I went to bed soon. after.

You would have to say "soon after". "Soon" means that you are expecting something in a short amount of time. "Soon after" means after a short amount of time, something happened.

Yesterday was a very lazily day.

I will work hard next daytomorrow!

You would have needed to say:
I will work hard the next day!
But I changed it to tomorrow, since people would typically say "tomorrow" instead of "the next day".