When to use pronouns in Japanese?

For example, if I'm writing something and I say "watashi wa A desu" then I don't need to use pronouns for the rest of the text, right, since it's assumed that I'm talking about myself through the whole thing ...?

If I say "watashi wa A desu" and then say "nihongo wa heta desu", I don't need to put "watahsi wa" before "nihongo", do I?

correct. you can abbreviate it.

It is more normal to say .
"A desu. watashi wa nihongo ga heta desu"

What if you were saying "My name is A. I'm studying Japanese, but my Japanese is really bad"?

"What if you were saying "My name is A. I'm studying Japanese, but my Japanese is really bad"?
No need to say it, it will be obvious..

The Japanese person you are talking to will then feel obliged to reply "not at all, you are doing fine.."

If you mentioned about who in the first line and if you are continuing to talk about same person, you do not have to keep saying "watashi ha " cuz it is obvious from the speech who you are talking about. If I take your example "Watashi ha A desu. Nihongo wa heta desu. " It is obvious you are talking about "watashi=I" in first line then you mean second line is also about "watashi=I". When you change the topic of the sentence and the topic about is other person then you mention about pronouns. For Example. Watashi ha A desu. Nihongo wa heta desu. Oscar wa watashi no pet no neko(cat) desu. Hirune(having nap) ga daisuki(love verymuch) desu.

Last sentence is obvious about what this sentence mentioned about from third sentence, so you can skip the pronouns"Oscar"