could you please explain this sentence to me "The trade war is already hurting US companies in China". As I know "already" is often used with the present perfect or past perfect. Why they use it with the present continuous?
You say "already" is "often" used with the present perfect or past perfect, so you yourself already know apparently that it's not always used with those tenses. You can use "already" with other tenses; the present continuous is one of them. So is the present, I did it myself in this paragraph.
You use "already" to say that we might think Verb will only be true in the future, but no, Verb is true in the present. This is the case whether Verb is an action that was completed in the past ("I've already done my homework" -> the doing of the homework happened in the past, but we are talking about the present state of the homework being finished, when you might have thought it would only be finished in the future), or one that is currently happening (such as the trade war already hurting US industry, when we might have thought it would start hurting US industry later).