A long time ago, There were three little pigs.

The first letter at the beginning of a sentence should be capitalized.
But the first letter after the comma should be lowercase.

A long time ago, there were three little pigs.

They were brothers.

OK!

They lived on a small hill with their mother.

OK!

The youngest pig is named Joe.

OK!

The second one is named Tim.

OK!

The eldest one is named Tom.

OK!

They has lived with their mum for years.

{They} are plural, so we use {have} here.

They have lived with their mum for years.

But day by day, they grew up.

OK!

The house were too small to live.

{The house } is third person singular, so we use {was} here.

The house was too small to live.

One day, their mother kicked them out of the house.

OK!

They should built their own house.

{Should} is a modal verb that should be followed by the original form of the verb.

They should build their own house.

Joe liked to sleep he was a lazy pig.

We need to use {and} to connect two simple sentences.

Joe liked to sleep and he was a lazy pig.

He used grass to build his house.

OK!

Tim was lazy too he liked to eat, he used many sticks to build his house.

We need to use {and} to connect two simple sentences.
If this sentence is too long, please break it into multiple simple sentences

Tim was lazy too and he liked to eat. He used many sticks to build his house.

Tom was a hard-working pig he liked to work.

We need to use {and} to connect two simple sentences.

Tom was a hard-working pig and he liked to work.

He used many bricks to build house.

OK!

for his house, he has not slept for 2 days.

The first letter at the beginning of a sentence should be capitalized.

For his house, he has not slept for 2 days.

Many days later a hungry wolf arrived.

OK!

He saw their houses.

OK!

The wolf has not eaten for three days.

OK!

He wanted to eat pigs.

OK!

Tom staggered.

OK!

He had a splandid idea.

Watch your spelling. {splendid}

He had a splendid idea.

He put on the fire.

OK!

The wolf ran to the first house.

OK!

He blew down and ate Joe.

OK!

He ran to the second house.

OK!

He blew down and ate Tim.

OK!

He ran to last house but the house was too stable to blow down.

OK!

He decided to climb up to the roof and jumped in to chimney.

We use {into} to indicate entering.

He decided to climb up to the roof and jumped into the chimney.

But the fire was burning.

OK!

He died in the fire.

OK!