Good work! I'll take this one sentence at a time.

In the first sentence, your speech pattern is a little unusual but it sounds whimsical so it is okay. The only problem I see is, you used the word little twice very close together (in yellow). In English, this is very unusual, and in writing seems lazy. I would use the word "small" instead of the second "little"

I am so sleepy now but I am also a little sleepless, which is a little paradox. 

I am so sleepy now but I am also a little sleepless, which is a small paradox. 

In the second sentence, there is one thing. Instead of saying "leave home for work" it should be "left home for work". This is because you are talking about the past. 

I sleep in my friend's bed these days because I leave home for work.

I sleep in my friend's bed these days because I left home for work.

For the third sentence, you have the right idea but the wrong execution. Instead of "have got used to" it should be "has gotten used to". This because you are talking about the past again. 

I am kind of person who have got used to their own bed.

I am kind of person who has gotten used to their own bed.

In the fourth sentence, there is one mistake, and one enhancement. You need a capital "I" after "but" (red) and instead of saying "go to sleep", I'd say "fall asleep". In English, to "fall asleep" represents going from being awake to asleep. Saying "go to sleep" is like saying "go to bed". A good way of thinking about it is "you go to bed, to fall asleep". 

So I toss and turn all the night but i can't go to sleep.

So I toss and turn all the night but I can't fall asleep.

The fifth sentence has two spelling mistakes. "I" by itself is supposed to be uppercase, and instead of "fell", it should be "feel".

Whatever sleeping position, i still fell uncomfortable.

Whatever sleeping position, I still feel uncomfortable.

For the sixth sentence, instead of saying "In the result" it sounds better to say "As a result". The second mistake is the lowercase "I". For the last part of this sentence, it isn't wrong to use "in" but you would have to add the word "the" making it "in the daytime". However, if I was writing the sentence, I would use the word "during" instead of "in".

In the result, i am so sleepy and exhausted in daytime.

As a result, I am so sleepy and exhausted in the daytime.

As a result, I am so sleepy and exhausted during the daytime.

The seventh sentence is largely good except for one spelling mistake, it should be "snowing". 

In addition, it is sonwing these days and so chilly outside. 

In addition, it is snowing these days and so chilly outside.

For your last sentence, there are 2 spelling mistakes and one enhancement. The word "I" needs to uppercase, and you don't need the word more.

Oh, i hope i will get better and the weather gets more warmer.

Oh, I hope I will get better and the weather gets warmer.

I hope this helps! If you have questions please ask.