“The US dollar is going up/coming up.” “The US dollar is going down/coming down.” Do these sound natural to mean “the value of the US dollar is increasing or dropping”?

These phrases would make sense to me in casual conversation. For maximum clarity you might want to give more details and say "The *value* of the US dollar is going up/down/rising/dropping", or even "The US dollar is weakening against the British Pound". If you say the dollar is changing value, your partner's next question may be "changing in relation to what?". But there is nothing immediately wrong with those phrases.

Not really, in my opinion. You could say “the *value* of the US dollar is going up/down” I think I would use “going up/down” over “coming up/down” in most scenarios. One example I can think of where I’d use “coming down” is if the value suddenly increased and was expected to quickly decrease back to ‘normal’

Rise or strengthen/get stronger and fall or soften or weaken/get weaker are common ways of talking about a currency. I recommend browsing finance/economy sites to get a sense of the usage.

going up, coming down