Sentence by sentence: I think that some people don't want to have pets because they have little free time to take care of them. (In English you put a comma in front of "that" less often than you would in front of что in Russian) I'm not sure the exact idea you are trying to get across in the second sentence? Forget like, after the pets die? When the people are on vacation? I wouldn't keep exotic pets like spider or snakes because I'm afraid of them. (More direct than your version, and "don't approve" sounds like you are making a judgment on someone else's life, which may not be intentional). I find it difficult to take care of pets, and they can also be expensive. Cats, dogs, hamsters, and parrots are very popular pets in my country (maybe "where I'm from" sounds a little more natural - or just "in Russia" if that is where you are from). In your email you told me about your sister's birthday party. Where was it held? How many people were there? Did everyone have fun? (These changes just sound more natural to me, but your version was fine too).

I would omit the first comma. Many people get very attached to their pets, then it is very difficult for people to forget their pets. (Can't use 'then' in the middle of the sentence that way. 'So' works nicely, or you can start a new sentence with 'Then') Cats, dogs, hamsters, parrots (This can be acceptable, but in normal speech you would include 'and' in the sequence) Was the birthday party existing? (I think you meant the word 'exciting.')

There *are* a few mistakes in your text: "I think, that some people don't want to have pets because people have little free time to take care of pets." should be "I think some people don't want to have pets because they have little free time to take care of them." "Many people get very attached to their pets, then it is very difficult for people to forget their pets." should be "Many people get very attached to their pets, so it is very difficult for people to forget them." "I don't approve of keeping exotic pets like spiders or snakes, because I'm afraid of them. I find it difficult to take care of them, they can also be expensive." should be "I don't approve of keeping exotic pets like spiders or snakes because I'm afraid of them. I also find them difficult to take care of, and they can be expensive." "Cats, dogs, hamsters, parrots are very popular pets in my country." should be "Cats, dogs, hamsters, and parrots are very popular pets in my country." The sentence "In your e-mail you told me about sister's birthday party. Where was the birthday party held? How many guests were at the birthday party? Was the birthday party existing?" is a bit confusing. "In your e-mail you told me about sister's birthday party" should be ""In your e-mail you told me about your sister's birthday party." Also, it's not clear what you're asking. Are you asking if the birthday party actually happened? If you're asking if the birthday party actually happened, you could ask "Did your sister's birthday party happen?"

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