你好！【我姓___，我叫____。】/ 【我姓名_____。】 我十六岁，和我现在在十年级学生。
English grammar and Chinese grammar might have difference, but basic writing rules still apply to both. For example, "My name is XX, I am 16 years old, I am a grade ten student" may not be grammatically incorrect, but any decent writer would know to avoid repeating the subject too many times within a series of related sentences and write "My name is XX, I am a 16 years old grade ten student"
"和"(and) in "我现在在十年级学生" is not needed, and this sentence has no verb (I now at ten grade student). The correct structure is " I am a ten grade student" (现在是十年级学生)
Again, you use 我 to start two sentences in a row, but the subject in the first sentence is "我的家" and the subject in the second sentence is "我", therefore not omitting 我 is correct.
If you wrote "我住在美国，(我)是_____高中的学生。" then the subject in the first and second sentence would both be "我". In this case, you should omit the second 我 in the second sentence.
我要 sounds like a command, better use 我会让 (I would make)
在进步中 (is improving) is incorrect in this context, you cannot "make your Chinese is improving" ; you can only make "your Chinese to improve"
Use final particle '的' to indicate the sentence is completed and add tone of 'keenness' in your speech
"我有妈妈，爸爸" Everyone have father and mother. Your sentence didn't indicate the fact that you are living with them as one family. What you really wanted to say was :"In my family there's my father and mother..."
"我的家人也有一只狗" means "having a dog, same as everyone else"; I think you wanted to say "还" (also) meaning "in addition of all the human family member, we also have a pet dog"
add "有点" (a little) before "害羞" So that people wouldn't think you are 'too shy'
add "仍然" to indicate despite the fact that you are shy, you still are willing to reach out to people.
"(也)想要交到更多的朋友。" is better than "(而且)我想要交到更多的朋友。"
You already used 乐于 in the previous sentence, use 喜欢 this time so you wouldn't repeat 乐于 too soon.
Beside, '乐于' could also mean "willing to" ; while "喜欢" can only mean "like to" . I think it is want you wanted to say.
In this casual format, use "也很" is better, because it is more colloquial than 并且,