Letter N°2

Since you are composing an email, this would more accurately be called "Email N°2." Emails are not considered letters.

In my free time, I usually play video games or also ride my bike in my gyarden.

You could also say "... and also ride...", it carries the same meaning

There is a difference in meaning between "garden" and "yard." Gardens are typically where you deliberately grow plants (flowers, vegetables, etc.), and tend to be dense with those plants. Yard means it is part of your property, but are typically mostly "flat" with grass.

I don’t have any problemconflicts.

This is to mean that you don't have any other plans to prevent you from spending time with your friend.

But in the morning I’m not free, II just have a medical visit in a clinical not far awa nearby.

More natural.

Not far away = nearby

Clinic = noun
Clinical = adjective

For me, going shopping in the city centre or going to a park doesn't change anything, fwould be the same. For this, you can ask Lily, she is very reliable.

Sorry, but in the evening I’m going to have a dinner with my Ggrandparents in the centrmiddle of a (country?) not far away.

No need to capitalize "grandparents."

"Middle" is a synonym for center/centre and is more natural in this sentence. In general, it is a more casual word.

Do you mean country here or did you mean a city/town?
Countries: France, Canada, Australia, Italy, Saudi Arabia...
Cities: London, Manchester, Vancouver, Toronto, Los Angeles, Seattle, Dallas...

I hope you answer me as soon as possible.

There is no error here, but it can be interpreted as a little pushy/aggressive.
If your intention is to say that you would be happy/excited to hear from them soon in a softer way, you might instead say: "I hope to hear from you soon!"

Thank you for reading ¶
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Goodbye ¶
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Take care,¶
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Edoardo

"Goodbye" is not really used over email; it is for in-person.

There are a lot of ways to sign off an email, but "take care" is a safe option for friends and family.