2. In Japan, you have tomust take off your shoes wheoff upon entering a house.

"Take your shoes off" is more fluent sounding. The other 'corrections' like "must" and "upon" are just alternative choices.
「take your shoes off」がもっと流暢だと思います。「must」と「upon」は代替だけです。
（下手な日本語すみません！）

Otherwise, pets such as dogs and cats mightany cats or dogs in the house might try to bite you.

In English, the order "cats and dogs" is usually used over "dogs and cats".

"pets such as dogs and cats" feels a bit unnatural. Perhaps it's something to do with the fact that cats and dogs are so common pets, and that it feels strange that you would mention "pets" in general but then specify only these two very common ones. I can't pinpoint exactly why, I'm afraid, but I've corrected it to something that feels natural in English.

My watch had been planted bombfitted with anti-theft measures.

Typically, bombs are planted somewhere, rather than somewhere being planted with bombs.

"Bomb" makes me think of a very powerful explosive, which would be unlikely to fit in a watch (unless in a James Bond movie or similar).
I've just replaced "bombs" with the more general "anti-theft measures".

I haved previously been on the fence butabout using them, but now I turned ithem on.

The tense of "have been" didn't really fit with the rest of the paragraph.

It was unclear what you were "on the fence" about.

"turn on" implies that something can consistently be either on or off. ("activate" can be used with pretty much the same meaning.) E.g. a bomb wouldn't usually be "turned on", since an explosion is pretty instantaneous. A bomb would instead be "detonated".

For the "them", I'm referring to the "measures" in my correction of the previous sentence.

Hot pepper powderTear gas bombs exploded and they burst into tears on the spot.

"Hot pepper powder" sounds like you would literally have a ground powder of hot (=spicy) pepper, or hot (=heated) ground powder from pepper. I suspect that you actually mean "tear gas".

Even though "burst into tears" should make sense here, it actually makes me picture an emotional response, when I suspect you just want to describe that the explosion (or the gas etc) immediately triggered it.

It took off 5five years off my age.

"I" didn't do anything to the age. Rather "the treatment" did.

Small numbers are often written with letters, rather than the numbers themselves ("five" vs "5").

5. I have to skip breakfast.

If you just write "I skip breakfast.", I understand this as a general statement, i.e. "I never eat breakfast". This may be true, but in order to fit with the next sentence, it needs something more to show that you're talking about the current moment specifically.

I'm taking off to work.

The original reads as "I'm taking off (=leaving) for the purpose of working." I presume that you mean that you're going to a specific physical location (i.e. "work" as a noun, rather than "work" as a verb).

This and the previous sentence could be rewritten together in a few ways, e.g:
"I'm going to skip breakfast. I have to go to work."
"I don't have time for breakfast today. I have to get to work."
"No time for breakfast, I have to run."

There's probably a way to write this in a natural way with "take off", but nothing springs to mind right now.

I tried to take offlose 3kg but screwed it up.

"take off" isn't used when referring to the steady process of losing weight.

I put on 3kg on the contrarynstead, I gained 3kg.

You could still use "put on 3kg" here, I just changed it to "gained 3kg" for a bit a variation, since you'd already used "put on".

7. My neighborhood is under constructionThere's a lot of construction work going on in my neighborhood.

The original implies that the whole neighbourhood is under construction (=hasn't been built yet).

The bigAll the noise is taking my attention off fromdisturbing me when I'm trying to studying English.

You can't "take off" attention.

9. She stayed silent for a while but took offhen started to talk about that sad accident.

"take off" isn't used in the sense of "start doing". I've tried to correct it to how I understand your Japanese sentence. I may not have managed to do it justice, but at least the corrected sentence sounds ok in English.

"sad" isn't particularly strong when talking about accidents. It's not necessarily wrong, and maybe it's appropriate for what you're trying to say, but some stronger adjectives that would also work here are: tragic, terrible, horrible.

We can't stand this event anymore.

An "event" is limited in time, such as a sporting event or a party. My first thought with the original sentence was that you're at some party or formal event, and just want to leave, which doesn't fit with the next sentence. On the other hand, you might be on the first day of some week-long event, in which case it would work. You might also be trying to express something more along the lines of "situation".

Let's stake off to carry out a signature collection campaign to ask the government to stoprt a petition against it.

"take off" isn't used to mean "start doing something"

"signature collection campaign" would be understood, but sounds quite unnatural. "petition" is typically used instead.

"to ask the government to stop it" sounds quite long. It could still be used to make it explicit, but usually it would sound more natural with just "a petition to stop it".

Depending on what "it" refers to, the action ("to" + verb) will change. For an event, you could indeed use "stop", but "cancel" or "shut down" might be better.

"against it" is less specific, but more general.

Two guys tried to take a girl off from her mother oin the street.

That musical suddenly took off because a main actress got injured atin a car accident.

Note the difference between "in a car accident" and "at a car accident". The following two examples work better with e.g. a paramedic:

"The paramedic was injured in/at a car accident."

"in": the paramedic was in the car, and was directly injured

"at": the paramedic was present at the site of the accident, and was injured as they were carrying out their duties

I'm taking off fromtime off work on Thursday to take my aunt around Tokyo.

But my aunt is veryalways nagging at me.

"nagging" on its own like this sounds unnatural. It would usually be used like this: "nagging"+noun, e.g. "a nagging pain" or "a nagging voice".

Shall we stake off fromrt with self-introductions?

I might look cute but I'm a big eater.

The corrected sentence is grammatically ok, but sounds a bit strange.
In general, I have to say that I would be surprised to hear someone say that they were "cute" or "a big eater" in their own self-introduction.

In English, "cute" isn't normally associated with how much or how little you eat, so it sounds strange to put forward being "a big eater" as a contrast. In this sense, it would be more natural to say something like "I might look cute but I have a black belt in judo."

Or to contrast with the eating, you could just refer to physical size: "I might be small but I'm a big eater."

1. I'm takgoing toff for Osaka on a business trip.

"taking of"is too informal for this contect. You can say to a friend when you are about to leave their house "I'm taking off."

3. They took my wrist watch off and ran away with it.

Not a great example. You wouldn't use "take off" to describe robbers grabbing your watch. You'd say something like "they pulled my wrist watch off". You can say "I always take my wrist watch off before I go to bed."

My watch had been planted bombs.

Doesn't make sense.

I have been on the fence but I turned it on.

Doesn't make sense.

Hot pepper powder bombs exploded and they robbers burst into tears on the spot.

You need to specify who "they" is. Otherwise the sentence is confusing.

It took off 5five years.

It standing in for "the facial treatment". Numbers less than 10 should be written out.

5. I'm skipping breakfast.

I'm taking off tofor work.

Normally (at least where I live) "I'm taking off" is said as a complete sentence. If you are going to continue the sentence to say where you are going, you use "going to" rather than "taking off".

I tried to take off 3kg but screwed it up.

It's not wrong but it is not something a native speaker would say. We would say "I tried to lose 3kg..." A better example would be something like "I'm hoping to take off three kilograms before my wedding."

I put on 3kg on the contraryinstead.

7. My neighborhood is under construction.

This implies that every building in your neighbourhood is currently under construction. You could say "There is a lot of construction work in my neighbourhood."

The biunceasing noise is taking my attention offaway from studying English.

"Big" is not used for noise. And that's not a good example of "taking off".

9. She stayed silent for a while but took off about that sad accident.

I've never seen "took off" used like that.

Let's take off to carry ou. We could start a signature collection campaign to ask the government to stop it.

"take off" used like this has to be a sentence on it's own.

Two guys tried to take a girl off from her mother on the street.

"take off from" used like this is wrong. You can say "the plane took off from New York airport".

That musical suddenly took off because aafter the main actress got injured atin a car accident.

There will only be one main actress, so "the" not "a". Otherwise you could something like "an important actress"

But my aunt is very naggingnags a lot.

Shall we take off from self-introduction?

This sentence doesn't work. You can't use "take off" this way. The sentence should be something like "Shall we start with everyone introducing themselves?"

I might look cute but I am a big eater.