The concept of Mutually Assured Destruction is usually more than enough to deter would-be combatants from deploying missiles.
You are making a series of assumptions out of thin air.
Many people (Democrats included) have questions about Trump's ability to think rationally about military matters, this isn't limited to cases where Mutually Assured Destruction apply. He doesn't appear to understand why others are unwilling to use nuclear weapons. Hence this is a serious issue for many.
To suppose that North Korea is a serious threat to the USA is to seriously underestimate the military power of the United States. It is the same sort of thinking that has deluded thousands into thinking that Russia is somehow a superpower.
N. Korea's nukes are a threat to about 37,000 US servicemen in Korea, they are also a threat to US Navy ships in the region, but the issue being discussed isn't North Korea's use of nukes, the discussion is about Trump's potential use of nukes. Trump's false assumptions about military matters give potential for him to paint himself (the US) into a corner where the only facesaving way out may appear to Trump to be use the nukes we have in our inventory.
China exerts a large amount of control over North Korea.
Probably not as much as you think; but again, it's Trump we are concerned about here, not North Korea.
There is little reason to suggest, given the context, that they are thinking of anyplace other than Russia.
To begin, you're wrong in thinking that the issue of not trusting Trump with nukes revolves around Russia. It revolves around Trump; where he might decide to go off the reservation is unknowable, but it surely isn't limited to Russia.
Saying that "Putin would walk all over Trump" suggests something completely different to the idea that Trump is somehow working for, or is influenced by, Putin.
Trump clearly thinks that Putin is his alter ego or soulmate or some such thing. Putin will gladly encourage that kind of thinking in Trump. Until the inevitable crunch, and Trump is out of his league where it comes to Putin, his Dad's attorneys can't help him there.
There's a reason why "confidence men" are called confidence men.