First Issue:
Because of the destructive nature of nuclear weapons, obviously I would rather no one have them. But in reality, the cat is already in the bag and because competing nation states have nuclear weapons in their possession, they would likely never relinquish them all because of the fear the another nation state was lying and held some back in their possession. As for unstable people getting access that are in government, you should research the current safeguard we have in place so that only a few men have the ability to arm and launch nuclear weapons and those people are vetting quite harshly and are given continual physiological evaluations you mental stability.
And we can't have everyone agree not to own nukes and police each other for it, while having a private defense force in charge of keeping out outsiders, whom we entrust with owning a nuke for deterrent reasons because...?
Second Issue:
That is an assumption they would be found out, if you know what your doing you could be covert about it. There are small tactical nukes are are as small as a suitcase. Also in your example, you gave a price tag so I had to assume that person was purchasing it from somewhere not building their own. Building your own it a whole other ball game and is very hard to mask because of the equipment needed, power consumption and raw materials. You really need to understand, covert action happens everyday and it is effective more times than not. History is filled with examples (that we know of) and it is only decades if not longer that we find out, or we seize records and read about them after the fact.
If someone sold someone a nuke, then they know whom they sold the nuke to. If one of those nukes does go off, either whoever sold the nuke will have told everyone whom they sold that one to, or they would be held responsible. And yes, the price tag is the estimates cost of a nuke on the black market. But, regardless, why is covert action and black market suitcase nukes a problem in anarchy state, but not a problem in government controlled state? What makes people with a government paycheck different or better than people with a private company paycheck?