I'm very excited because recently I built a barbecue area at homemy house my recently and it looks great! I'm very excited because I built a barbecue area at my house my recently and it looks great!

The original placement of "recently" is correct, but it sounds more natural to put afterwards.

I'm very excited because recently II just built a barbecue area at home and it looks great! I'm very excited because I just built a barbecue area and it looks great!

Hi Joe, I'm writing you this message to invite you to my birthday celebration on this Saturday. Hi Joe, I'm writing you this message to invite you to my birthday celebration this Saturday.

If this was to be sent as an e-mail rather than by text ("message" is often used for texting) or physical mail, it might be more appropriate to say "I'm sending you this email" or "I'm writing you this invite".

Hi Joe, I'm writing you this message to invite you to my birthday celebration on this Saturday. Hi Joe, I'm writing this message to invite you to my birthday celebration this Saturday.

"You" isn't necessary here Either "on Saturday" or "this Saturday" is correct, but it can't be both

Hi Joe, I’m writing you this message to invite you to my birthday celebration onparty this Saturday. Hi Joe, I’m writing you this message to invite you to my birthday party this Saturday.

I pretendlan to makhave a barbecue at home with some friends and close family. I plan to have a barbecue at home with some friends and close family.

Not quite sure what you mean here by "pretend". If you are writing about what you will do at your birthday celebration, you could replace it with "plan". When referring to holding a social barbecue event, it is more natural to say "have a barbecue".

I pretendlan to makhave a barbecue at home with some friends and close family. I plan to have a barbecue at home with some friends and close family.

"Have a barbecue" is the correct term

I pretend to make a will barbecue at home with some friends and close family. I will barbecue at home with some friends and close family.

I'm looking forward to seeing you and spending a relaxing time together! I'm looking forward to seeing you and spending a relaxing time together!

Could also say "having a relaxing time".

I'm looking forward to seeing you and spending a relaxing time! I'm looking forward to seeing you and spending a relaxing time!

"Relax" is the verb, "relaxing" is the adjective

See ya , Claudio. See ya, Claudio.

See ya !/, Claudio. See ya!/, Claudio

Punctuation after the closing (for this context an exclamation point would work best, but a comma is also fine) No punctuation after the name

See ya ou soon, Claudio. See you soon, Claudio

I hope you are free and you can come to drink some beer with me! I hope you are free and you can come drink some beer with me!

A more natural way to phrase it might be "and you can come have some/a beer with me".

I hope you are free and you can come to drink some beer with me! I hope you are free and can come to drink some beer with me!

You don't need to add another "you" here, since both verbs are being done by "you"

I hope that you are free and you can come to drink some beer with me! I hope that you are free and you can come to drink some beer with me!

By the way, I remember that you told me you were dating with your workmate Maria. By the way, I remember that you told me you were dating your workmate Maria.

"that" can be omitted here to make it sound smoother, however it is also fine to keep it.

By the way, I remember that you told me you were dating with your workmate Maria. By the way, I remember that you told me you were dating your workmate Maria.

"Dating [person]" no with

By the way, I remember that you told me you were dating with your coworkmater Maria. By the way, I remember that you told me you were dating your coworker Maria.

It would be a good opportunity to get to know her and introduce her to my wife, they might get along well! It would be a good opportunity to get to know her and introduce her to my wife, they might get along well!

Although the sentence is correct, replacing "it would be a good opportunity" with "This celebration would be a good opportunity" would make it clearer to the reader what "it" is in this situation.

It would be a good opportunity to knowmeet her and introduce her to my wife, they might get along well! It would be a good opportunity to meet her and introduce her to my wife, they might get along well!

"Meet" or "get to know" is the correct verb here instead of "know"

It would be a good opportunity to knowmeet her and introduce her to my wife, t. They might get along well! It would be a good opportunity to meet her and introduce her to my wife. They might get along well!