My spaghetti is getting cold.
My iced tea is getting warm.
I’d usually use cool when (e.g.) something is still too hot to eat and therefore I need to allow it to cool off before eating.
My drink is getting hot => I don’t think I’d usually use the word hot unless possibly a drink is sitting in direct, prolonged sunlight (and the temperature of the drink is becoming very high) or unless I’m actually heating up a drink that is supposed to be hot.
It’s a commonplace to say: Hurry up and eat because the food is getting cold. In other words I’ve cooked the food and I don’t want you to miss the chance of it becoming cold as it will not taste as good. You are asking people to eat and not let it get cold.
Cool on the other hand is what you would like the soup to become because it’s too hot to eat at the moment as in: Let it cool down/let it get cool before you attempt to eat it because otherwise you will burn yourself.
I’m not sure I really understand your last question.
With reference to cold:
For me it’s “standard” to say that your hot meal is getting cold if it is left uneaten for too long after being cooked and served. It’s not my invention or decision to say it that way, it’s just the usual expression.