Can anything exist outside of space-time?
The following video also asks a question I would like to know the answer to.
First of all, that guy keeps on saying he understands physics and string theory when he clearly doesn't. Nonetheless....
The question the video asks is "What is around the universe?" which is related to "Can anything exist outside of space-time?"
Picture a balloon that's blown up a little bit. Get out your imaginary sharpie and start drawing some stuff on the surface of the balloon - throw in some galaxies, some stars, some quasars, some planets, and maybe some space-faring lemurs for fun. Now, this part is important: remember that you are drawing on the surface
of the balloon. The volume of the balloon, ie, all the air inside and outside the balloon, is totally irrelevant to our discussion. If you've read flatland then imagine the balloon-surface-dwellers are of a similar species to the squares and triangles of flatland - they only comprehend space in 2 dimensions, not our 3.
For the next step in our picture, you're going to have to pretend that the surface
of the balloon represents the entirety of reality. The volume inside and outside doesn't exist, because it's not reality (although it clearly does exist in our mental picture.) Once you've got that firmly fixed in your head, inflate the balloon a bit. What happens?
Sure, the volume
of the balloon expands into pre-existing volume outside of the balloon. However, the surface
of the balloon gets bigger without expanding into any other pre-existing surface. It doesn't inflate or expand or explode into some other pre-existing surface, it just has a larger extra-dimensional space in which to exist. There was no empty balloon-surface into which our space dwelling lemurs expanded into, rather, the surface between the galaxies and quasars and what-not simply got bigger.
Now map that picture back into our glorious 3 dimensional world. That's how the expansion of the universe worked (and is working.) There is no empty space outside of the universe. The universe is, by definition, the entirety of space - empty and occupied - and it's getting bigger.
Your related question, "Can anything exist outside of space-time", is another question. The number "17" clearly exists outside of space-time. You can point to 17 trees, or 17 space-faring lemurs (actually truth be told, you'd be hard pressed to find any space-faring lemurs, let alone 17 of them), but you can't point to the number 17. It exists, because we all know the thing we're talking about when I say "17" (the number one greater than 16. The 7th prime. The number you get when you halve 34.), but you can't point to it anywhere within space-time - so clearly it exists outside of space-time. That's probably not the question you were looking to get answered, though.