Why does this sentence not need need रहना? Would वे बच्चे खड़ रहे हैं also work? Or is the verb “to stand” an exception?
You guessed it right. "To stand" works differently in Hindi: it's expressed as an action in English but as a state in Hindi, and hence the word used in the latter case is not a verb but an adjective. वे बच्चे खड़े हैं is grammatically identical to वे बच्चे X हैं, where X is any other adjective. I've posted explanations on this topic in a bit more detail here and here, in case you wanna know more about the difference.