I felt secure when falling asleep with it, and it wa. It is a soft, fluffy and cute penguin.

Reads better as two sentences here since there's only a loose relation between the clauses.

I changed the second sentence present tense, because your later sentences say you still have the stuffed toy, while talking about it in the past tense implies it's gone or you no longer have it. You could also change the first sentence to "I feel secure when falling asleep with it", if you still sleep with it.

When I was upset or going through hard times, I willould hug it as a way to comfort myself.

"will" -> "would" to match the past tense of the first clause

It makes me feelt better.

It is important for me because I treats it as my best friend.

My mom told me that no adult will hugs stuffed toys duringwhen sleeping, I will keep hugging mine though.

I felt securafe when falling asleep with it, and it was a soft, fluffy and cutecute and fluffy penguin.

secure is fine, but i feel safe is a little more natural!
same with the adjectives, changing the order sounds a bit better

When I was upset or going through hard times, I willould hug it as a way to comfort myself.

would is will in the past tense :)

It makesde me feelt better.

It iwas important for me because I treatsed it as my best friend.

since you're talking about the past, i changed some sentences to be in the past.
if you want to talk about now, you say 'I treat it' as my best friend

My mom told me that no adult would still hug a stuffed toys during while sleep,ing. I will keep hugging, though.

we use 'would' instead of will because what your mom said is just a future guess, not something definite about the future. and you don't need to repeat keep hugging again, we already know because you mentioned it in the previous sentence.

When I was upset or going through hard times, I willould hug it as a way to comfort myself.

This sentence needs the conditional “would” in place of “will”

It makesde me feelt better.

Past tense

It iwas important for me because I treatsed it as my best friend.

My mom told me that no adult willould hug stuffed toys during sleep, I willould keep hugging though.

When I was upset or going through hard times, I will hugged it as a way to comfort myself.

will hug = future
hugged = past

It makesde me felt better.

makes = present
made = past

It iwas important for me because I treatsed it as my best friend.

is = present
was = past

My mom told me that no adult will hugs a stuffed toys during when they sleep, but I will keep hugging thoughmine.