I wrote "the kid has the cold" as in "the kid has the cold [that has been going around]" - would there be another way to say this?
Yes. As far as I know very few languages use the word "cold" to refer to being sick.
verkouden is the word for the common cold in Dutch from what I can tell.
Even English doesn't use them in the same way: he is cold versus he has a cold (or perhaps he caught cold).
In Dutch, for the words 'warm' en 'koud' you say:
[pronoun] [hebben conjugation] het warm/koud.
Its just a quirk in the grammar. So one says:
Ik heb het koud. Wij hebben het warm.
For what I know at least all of Romantic languages have that form of saying one is cold/warm. Correct me if I'm wrong.
You mean romanic? Latin based languages do use the same without the last pronoun. In Portuguese you say "The child has cold".
That would literally mean that the child is cold to the touch. This sentence means that the child feels cold.
When you use the adjective as a noun, the word does not get an "-e". Like "This is an empty plate" versus "The plate is empty". On the second sentence, empty is used as a noun.
His pronounciation is wrong. In the end it seems like he is saying "couch"
Depends on the context if the child had/has a cold drink rather than a hot drink yes. But if you are talking about temperature of the child no it would not make sense. Has means present ownership of something. In English the way of saying this would be" The child is cold". If the child was ill the child has a cold. But this is Dutch not English so iv learnt not to question the grammar and just get on with learning it literally.