Do international institutions such as the UN weaken the power of nation-states?

Answer 1

Short answer: only when those nations-states choose to go along with UN policy.

Long answer:

There is no mechanism for most international organizations (IOs) to force a state to do something it doesn't want to do. The UN doesn't have a police force, and it certainly doesn't have "black helicopters", but it does serve as an important part of international relations.

The value of an IO is to set norms, sort of like a code of ethics, for how states should behave. Ideas can have a power all their own, and states that don't conform to these ideals can be exposed to pressure from outside. Perhaps the most important example of an international norm is the territorial integrity of states. Invading Another Country and taking their land is considered wrong, so one would expect countries to take steps to punish a state which tries to take territory. This doesn't always work, however, because it depends on states having the will to punish actors who break the norms. For example, the Russian invasion of Georgia (the country in Asia, not the American state) in 2008 went largely unanswered, because very few people (and no governments) were willing to risk war with a nuclear-armed state over relatively weak Georgia. On the other hand, the UN authorized the use of force by NATO and the Arab League in Libya in 2011, because leaders were willing to take on the Gaddafi regime.

Generally speaking, a state has to be acting badly, and built up a few enemies before an international organization will take action. Even if a country is sanctioned by an IO, it probably won't mean anything if the target is strong enough (the US, Russia), or has powerful friends (Israel, which was subjected to UN arms embargoes in the past, but still received weapons from the US and France, or Syria, which has not been subject to the same UN action as Libya, in part because Russia and China have an interest in keeping Assad in power).

Basically, it depends. How good is the IO at convincing states that their norms are best for guiding how nations act? Do nations respect the IO? Are nations willing to back the IO up in enforcing norms? The power any given organization has depends on a multitude of factors.