Сайн уу Одгэрэлээ,

Here are my suggestions:

Энэ нэрийг миний монгол найз надад өгсөн. * What you initially wanted to say is actually, I think,  'Энэ нэрийг монгол найзаасаа авсан'. But I think the former is more natural in Mongolian.

Би японд амьдран суудаг.*'Би японд амьдран сууж байна' is totally correct. However, it implies that you are now living in Japan and previously you have lived somewhere else. 'Би японд амъдран суудаг.' says that you live in Japan.

Засуулах хүсэлтийг минь хүлээн авна уу. 'Засах' means in this context that you are willing to correct on your own, 'засуулах' means you are requesting someone else to correct yourself. I would say 'Миний алдааг засаж, монголоор зөвийг нь хэлж өгнө үү.', which literally means 'please correct me and tell me the correct Mongolian way.' And here again, you are mixing up 'ь' and 'ъ'. But don't worry at the beginning of your learning Mongolian language.

I hope it helps. 

анх монгол хэл

анх монголоор бичсэн нь

Миний нэрийг Одгэрэл гэдэг. Энэ нэрийг би монгол найзаасаа авсан. /Энэ нэрийг миний монгол найз өгсөн. Би Японд амьдардаг. Миний бичвэрийг засч өгөх хүсэлтийг минь хүлээн авна уу. 

The first text written in Mongolian/My first writing in Mongolian

My name is Odgerel. I took this name from my Mongolian friend. OR: This name was given by my Mongolian friend. I live in japan. Please accept my request to correct my writing. 

I think your writing is not bad. Good luck!