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My Favourite Pet is Rabbit

Animals are man’s best friend. I love all the animals but Rabbits are my favourite. I have a pet Rabbit. Her name is Zara and she is white in colour.
She was given to me as a gift by my uncle. She has beautiful red eyes and her paws have brown colour on the edges. She is intelligent and full of energy. I play with her when I go back home after school.
When I wake up in the morning, I go and spend time with her. I give her coriander leaves which she loves the most. Sometimes she comes to me before I wake up and hops on my bed. I take her to walk every morning.
I keep her in a basket which I decorated myself. Her ears are very sharp, she listens to us even when we whisper in night and comes to play with us. Sometimes, to tease her, I scare her with stuffed toys and she freezes herself out of fear. She looks so cute and innocent when she gets scared that I cuddle her a lot after that.
My Rabbit likes to dig in our backyard. When she makes a burrow, she hides herself init. She gets very happy when I find her in it. It is her favourite game.
She loves to eat fresh leafy vegetable and also carrots are her favourite. She enjoys rain a lot. Whenever it rains, she becomes very excited and goes out in the garden.
To keep her clean, I brush her every day and clip her nails in three days. My rabbit is my best friend and I love her the most.

В этом случае можно говорить he/him/his.

he конечно же. вы же говорите он (кролик) а не оно (кролик)
His name is или We called him

вообще нас учили писать про неодуш. придметы и про животных it
тогда its name is...

it. Thes (it) name is....

это в русском у нас принято как с животными, так и с людьми употреблять местоимения -он, она-...В английском немножко не так... У них с животными, не смотря на то, что они живые существа, употребляется местоимение "ит"(оно)... Но это общее правило! А так, свое домашнее животное, пол которого вы можете определить, можно назвать и с местоимения - он-она-. ошибкой не будет... Но вот какую-нибудь бродячую кошку или уличную собаку вы назовете "ит", так как не можете точно определить пол. Вот такие заморочки у английских товарищей.... Если вы хотите сказать, что вашу кошку зовут так-то, то начните предложение с фразы It name is...