Is it possible to turn the world from a Utopia to a Dystopia while still keeping a "Headquarters" that is utopian?
Absolutely. There's countless ways to make this work. One extreme example is with neural interconnects or drugs which let someone live in a Utopian world while the actual world crumbles beneath them. This is actually quite a common trope with VR
Would the Utopian world slowly diminish and eventually become fully Dystopian or do you think the life could keep up the small Utopia within?
This answer is dependent on countless details, so the answer would have to be "maybe." It's trivial to show an apparent utopia that crashes into a dystopia. On the other hand, it's also possible to make utopias that can exist forever, depending on your own individual definition of a utopia. For example, if your definition of utopia only includes the relative power of things, your utopia can shed energy and other resources into the outer dystopia while remaining utopian, so long as it sheds those things in a way which retains the balances your utopian definition calls for.
One particularly tenuous Utopia is one which is dependent on the possibility that something in the dystopia will eventually cure the demise of the utopia. As long as the utopia retains enough capabilities to reach out into the dystopia to find that cure when it appears, your utopia retains its status. Of course, whether this qualifies as a utopia depends entirely on you, as there is no defined "best world" that everyone agrees upon.
If the intelligent life overrun the current government etc, would it have enough technology to keep the world from becoming a dystopia?
Unknown. One of the great things about our world is we don't know whether what we create will be a dystopia or a utopia. We have to try it and see. Also, you get to define what a dystopia is, so you can make it as easy or as hard to become a dystopia as you please.