What a sweet notebook entry. I am surprised you haven't received any corrections yet! Your English is very is easy to understand but there are a few minor corrections I can help with to make it sounds more like a native speaker.

My Cat

I have a cat, and he is one a part of my family. He loves me and I love him too. Having a cat lets me know that for me, he is just a part of my life; but for him, I am the whole life of him  his whole life.

我家有一只猫，他是我们家庭的一员，他爱着我，我也爱着他。养只猫，让我知道，他只是陪我度过一段时间的宠物，而对于他，我是他的一生。

The My cat is pleased to accept all my emotions, such as angry anger, happiness, grief etc. He is pleased to  accepts the humid weather, my hugs and kisses, even though he does not like them. (Saying that someone is "pleased" and that they "like" something is too similar andsounded too contradictory so I took that part out. I think your cat probably likes your hugs and kisses though!)

猫咪欣然接受我的喜怒哀乐，接受潮湿的气候，接受我溺爱的拥抱和亲吻，虽然他可能并不喜欢。

I do not know what is the world in the world is like through a cat's eyes. Am I the closest relationship of him he has? I do not know whether my sorrows and joys are deserve his care more or not. How horrible about are disease and pains to a cat? Do all of he want is three meals and a warm nestIs ? Is a warm bed and three meals a day all that he wants?

我不知道猫咪眼中的世界是什么样的，我是他最亲近的人吗，我的悲喜值得大惊小怪吗，疾病和痛苦，对于一只猫有多可怕；是不是一日三餐，一个温暖的小窝对他来说已经足够了等等。

My cat likes to eat and sleep, and never wants to lose his weight. He enjoys his happiness and leisure life all day and night. Maybe he can thinking sth(Not sure about this word. Is it just a typo maybe?) just like the a human being (Using the word "person" here may sound more natural) does, and I do not know.

我的猫咪爱吃爱睡，从不减肥，舒服的享受着作为猫的快乐和闲适，或许他也可以像人类一样思考，只是我不知道。

Oh, my dear cat, it is so wonderful days these days are so wonderful. I live just because I have you. I feel warm in the winter and never feel lonely . You are the spring breeze to me and I will always take care of you until you are very old.

亲爱的猫咪，有你的日子真好，冬天不寒冷，我再也不孤单。你是我春天的微风，日夜不忘的牵挂，我习惯一如既往的爱你，直到你老去。

PS: I attached the Chinese here, maybe someone can give more advice or a better translation, thanks for all who cares about me.

I hope this helps you. I tried to leave as much of your original writing as possible to keep your intended meaning intact because it was so emotional. I just wanted to tell you that you should be proud that you used the word "lose" properly becaus<em>e </em>many native speakers can't do that! The words "loose" and "lose" have very different meanings in English, but are often confused my native speakers. Just something to watch out for when you are reading online! 

Also, this was my first time doing an edit like this so I hope it was okay and not too messy or confusing for you. Good luck in your studies!