When I commute to my office in aduring heavy rain, I usecarry an umbrella.

Therefore, I invented an solution for going out in a heavy rain.

Going Out in a Heavy Rain on Foot

I usually go to work on foot.

This is correct, but you can also say "by foot" instead of "on foot".

When I commute to my office in a heavy rain, I use an umbrella.

Therefore, I inventedcame up with an solution for going out in a heavy rain.

This is a more natural way to say this sentence.

I wear short pants and don't wear stockings.

By stockings, do you mean socks?

I put on stockings and change my short pantinto my trousers after I reach my office.

The solution has been working for more than 10 years.

You can also say:

"This has been an effective solution for me, for the past 10 years."

However, it hads a demerit.ownside (or a con).

I think the words "merit" and "demerit" are used differently in Japan. Here, you can use the words "downside" or "con" instead of "demerit".

I did thatt happened several days ago.

The bad smell came from bacteria that had grown, and the sandal's ordor was tooreally bad.

I washed them, but the ordor didn't disappear.

Going Out in a Heavy Rain on Foot (more natural: Going Out on Foot in Heavy Rain / Walking in Heavy Rain)

"A heavy rain" refers to an instance of heavy rain---a heavy rainfall. "A heavy rain" is possible here, but "heavy rain" is more natural, considering that this post isn't really about a specific instance of heavy rain, but about how you walk to work during heavy rain in general.

Going Out for a Walk in a Heavy Rain on Foot

The phrase "on foot" is used a lot by foreign speakers of English but it isn't something natives say very often. It's unusual that we would say "on foot" to describe how we are transiting between two places.

The below feels really unnatural to me:
"How are you going to get there?"
"On foot!"

Instead, we would use "I'll walk"
"How are you going to get there?"
"I'll walk"

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The expression "heavy rain" is not prefixed by the article "a".

For example:
"Look outside.. that's heavy rain!" (not "that is a heavy rain")

I usually go to work on foot.

The grammar here is fine. But, per previous comment, this is not something a native speaker would say.

"I usually walk to work" is what most people would say.

My house is not far from mythe office.

The grammar here is fine.

Since it is already clear that you are the subject of the sentence ("My house"), you do not need to use "my" for a second time.

In English, it's very common to simply say "the office" as opposed to either "my office" or "your office". For example:
-- "How long does it take you to get to the office?" is far more natural sounding than "How long does it take to get to your office?"
-- "It takes me 15 minutes to get to the office" is far more natural sounding than "It takes me 15 minutes to get to my office"

When I commute to mythe office in a heavy rain, I use an umbrella.

However, since it doesn't cover my lower legs, my trousers tend to get wet.

Excellent sentence!!!

Therefore, I invented an solution for going out in a heavy rain.

I wear short pants and don't wear stockings.

"Stockings" are a piece of clothing that women wear. You should Google the word "stockings" as I don't think this is what you mean :D

We refer to "short pants" as simply "shorts".
Example:
https://www.uniqlo.com/uk/en/men/bottoms/shorts?path=%2C%2C37374%2C

I put on stockings and change my short pantinto trousers after I reach mythe office.

Theis solution has been working for more than 10 years.

However, it hads a demeritrawback.

"demerit" is most commonly used in academic situations

I did than fact, I experienced it several days ago.

I tried to wear the plastic sandals last night, and found that they smelled very awful.

"very awful" is not incorrect grammar. But, in English, we tend not to use the word "very" with some words which already have a strong meaning.

"Awful" already means "very bad". So saying "very awful" just doesn't sound natural.

In school, we are taught not to overuse the word "very". Often, there is a better word out there to use than adding "very" to everything. Some English teachers love to hate on the word "very" because English speakers tend to over-use it.

If you want to emphasise even further than "awful", you can use a stronger word like "horrendous" or "horrific".

The bad smell bBacteria had grown andon the sandal's and their ordour was too badunbearable.

I washed them, but the ordour didn't disappear.

I hate theat musty smell.

Since you are referring to a specific smell (the 'damp' smell), you should use "that smell".