It's important to distinguish anarchy and anomie. The latter is the government collapse you mentioned, accompanied with lawlessness and lack of morals, while the former is simply lack of central overseeing authority (archism), for one reason or another.

The long term goal of anarchism is not destroying all governmental structures in one fell swoop, but rather gradually building communities based on liberty, solidarity and mutual help that don't require hierarchy or coercion to function.

Then those communities naturally take over governmental functions like protecting the people, the central government dissolves when it is no longer needed and the process doesn't harm anyone. No "new government" is created nor is necessary.

In terms of relationship between anarchism and libertarianism, I like to think of anarchists as a subset of libertarianists (since we all oppose authoritarianism fundamentally). I'll admit I'm not as familiar with other libertarian ideologies.