I’m thinking of writing a story about when I was ina middle school student.

"when I was in middle school" / "when I was a middle school student"

(The boiled eggs were peeled.)

We don't use peeled to describe eggs.

My father : said "Can I have a boiled egg?"

Me : I said "OK!

I’ll heat it up in the microwave."

Me : I said "It’s done!

yYou can eat it now."

Pieces of athe boiled egg were scattered in the kitchen.

"a boiled egg" would be some, previously unmentioned boiled egg. "the boiled egg" is used when it's the same egg we've just been talking about

Then, his top and bottom lips gotlips became red and swollen.

We'd just say "lips" unless there was a reason to talk about only one set of them.

Probably, I think my father probably also didn’t know about ithat.

The Egg Aand the Microwave

This makes it sound more like a story title where the "characters" are the egg and microwave
You can also say "Eggs and Microwaves" which emphasizes the relationship (egg + microwave = explosion) between eggs and microwaves more
"Egg and Microwave" sounds a little weird, if it's not plural it needs an article, "An egg and a Microwave"

I’m thinking of I'll writinge a story about when I was ina middle school student.

or "when I was in middle school"
I generally see「～ようと思う」translated as "I think I will," and that makes more sense here.
"I'm thinking of writing" is more like 「～ことを思っている・考えている」or 「～計画を思っている・考えている」

My father : Can I have a boiled egg?

No space between the speaker and the colon

Me : OK!

After that …

Me : It’s done!

Pieces of a boiled egg were scattered in the kitchen.

Okay, I'd suggest "scattered around the kitchen" to emphasize it being everywhere

Probably, I think my father probably also didn’t know about it.

"Probably," at the beginning sounds odd. We usually put "probably" somewhere inside the sentence, in front of the word it's supposed to most emphasize
"I probably think my father also didn't know it" <--- I may or may not think that (using probably this way about your own thoughts sounds unnatural)
"I think probably my father also didn't know it" <--- my father specifically may have not known. OK
"I think my father probably also didn't know it" <--- focusing on the "also not knowing" itself

I’m thinking of writing a story about when I was ina middle school student.

One day, my father came home from work and said, "I want to eat athe boiled egg on the table." (if there is only 1 egg on the table)¶
One day, my father came home from work and said, "I want to eat one of the boiled egg ats on the table." (if multiple eggs on the table)

(The boiled egg was peeled.)¶
(The boiled eggs were peeled.)¶

My father : Can I have athe boiled egg?

Here it is assumed that you either have already picked one of the egg out of all the eggs or there was only 1 egg. So your father would have requested for that specific egg.

When my father put it into his mouth and as it touched his teeth, it exploded.

Pieces of athe boiled egg were scattered in the kitchen.

Since we already know which egg it was, we use "the" instead of "a".

Then, his top and bottom lips gotbecame red and swollen.

Just more natural, got is not wrong though.

Probably, I think my father probably also didn’t know about it either.

When using "probably" at the start like that, it is usually in response to someone, and then followed by why you think that.
When making a statement, we usually put the probably in the middle of the sentence.

I’m so sorry. felt really bad. ¶
OR¶
Me: Dad, I'm so sorry. (if you are speaking to him)

I’m thinking of writing a story aboutwant to write about an incident that happened when I was ina middle school student. (or when I was in middle school)

It’s a true story that sounds like a lie, but it's true.

I think “it’s a true story that sounds like a lie” better depicts your original sentence

One day, my father came home from work and said, "I want to eat a boiled egg at the table."

My father : Can I have a boiled egg?

yYou can eat it now.

When my father put it to his mouth and it touched his teethAs soon as my fathers’ teeth came in contact with it, it exploded.

Pieces of a boiled egg were scattered in the kitchen.

At theat time, I didn't know I wasn’t supposed to heat it up in the microwave.

Probably, I think my father probably also didn’t know about it.

I’m so sorry. feel so bad

“I’m so sorry” is better when being said to the actual person you’re apologizing to