lLast year my pigeon died because of a concussion that happened after a blow by a penguin.

oOf course his death had no effect on biodiversity but in my heart it triggered an enormous typhoon started.

With the typhoon part it's completely optional - I just offered an alternative way to phrase it.

hHe became celestial but I was still on earth or even in subterraneanunderground.

The underground part is a little confusing still but the better choice would be to say "underground" instead of "in subterranean". Subterranean is an adjective so you would need a noun afterwards, and it would be used to describe something that kind of comes from below ground (there's probably a better definition but that's the best I can think of). For example, to talk about a creature that lives underground, you might say "subterranean creature".

sSo after that incident I decided on revenge.

I tried to arscommit arson on the penguin's house but sheit sought asylum for/from the monarch.

"Arson" is a noun, so you would need a verb ("commit") and you would need to say "on" afterwards as well.
"She" is fine, but typically you would say "it" when describing an animal unless you know the specific animal's gender, which is more often the case with pets (since you don't exactly seek to find out a random animal's gender, especially if it's out in the wild and you'll probably never be able to see/identify it again). This might also happen in zoos but I'm not 100% certain.
Choose "for" or "from" depending on the context. "For" would be more for if they're trying to protect the monarch, and "from" would be used if the monarch is protecting the penguin.

tThat time a fungus was the monarch.

I'm a little confused what you mean by this because fungus makes me think mushroom unless there's another meaning that I'm unaware of.
But if (since this is a really fictional story) you're saying that a mushroom, mould or some other meaning for fungus is the monarch it is technically correct, just a really obscure idea. I added in "a" and "the" because you would need "the" or some other article in front of monarch, and without "a" it sounds like fungus is the name of something. But if fungus is the name of the monarch, you would need to capitalise it ("Fungus" instead of "fungus").

I didn't get disheartened and instead got even got more angary and promised my pigeon that I would try so hard to overcome this obstacle in thise revenge plan.

Just sounds a little more natural like this.
"Even got" would be putting more of an emphasis on the act of getting angry, whereas "got even" puts more of an emphasis on the anger increasing. I added "Instead" to clarify the meaning a little.
"My pigeon that try so hard" would technically sound better as "my pigeon that tries so hard", but this would be saying that the pigeon is trying really hard, which isn't the case here. Instead, you're promising the pigeon that you will do something (but in the past tense which results in the sentence above).
Originally I thought that "overcome the penguin" might work well, but in my opinion it feels a little weird putting a living thing after "overcome". I think that obstacle would work better instead, and is also the more common thing to hear.
I chose "overcome this obstacle in the revenge plan" opposed to "overcome this obstacle in this revenge plan" because it just feels weird saying "this" twice and so close together in the same sentence, and the first "this" kind of gets the message across that you're talking about one specific revenge plan, along with the context provided by the previous sentences.

wWith woodpecker's agreementthe help of a woodpecker I decided to make a typhoon with sewer waters near the fungus's house, justand this torrential attack calmed his soul and mine.

Saying that a woodpecker agrees with you feels really weird. I think saying that the woodpecker helped would express the same idea to a certain degree, and also sounds a little more natural.
I don't really like how "waters" sounds since it's being used as a noun.
Instead of "just", I think that "and" would be a better choice, or you could just split it into two sentences.

he became celestial but I was still on earth or even in subterraneanunderground.

I tried to arson the penguin's house but she sought asylum forom the monarch.

I didn't get disheartened and even got more angary and promised my pigeon that I would try so hard to overcome out on top in this revenge plan.

with woodpecker's agreement I decided to make a typhoon with sewer waters near the fungus's house, just. this torrential attack calmed his soul and mine.