Why would -고 be used instead of -와 or -도 for the ending of 쓰고?
You add -고 at the end of the verb stem to connect two sentences with "and". Here, it just happens to be that the sentences are very short containing only a verb. The subject is the same for both.
Your connecting particles would be used for nouns for instance...
Because it's two verbs, not two nouns :)
My guess is because it's connecting verbs rather than objects but I'm not sure
is there any reason why I can't use "적고" instead of "쓰고"?
would 아이는 work as well? (instead of 아이가)
가 represent that we are talking about something specifically, while 는 is define we are talking about fact. For example
이 치타가 느립나다
This cheeta is slow.
Cheeta is/are fast.
Yes It also corrects!
I thought it would be 써고