A black hole is the same way. To some extent of course- I am not advocating the popular theory that black holes think and have built the pyramids. Wait, that's another one- sorry. Anyway, black holes curve spacetime to an infinite extent because they are not part of the universe. There is much evidence suggesting the existance of black holes. It is not impossible as it is sheathed in an event horizon. A black hole doesn't make any sense because it doesn't make any sense. To respond to one of your sentences: "This would mean that the escape velocity would be greater than 300,000km/sec meaning that light would not be able to escape." That is the very definition of a black hole- an object to which light cannot escape. Any such object is a black hole. This defeats a second sentence: "Also, why does a singularity have to have infinite density with no volume...". You need no infinite density. That is simply a 'by-product' of the property of a black hole- enormous gravity. The definition of a black hole is: an object to which light cannot escape. An example? Well, we all know the density of the sun. If you do some calculating, we can enlarge the sun, keeping its density the same, to the size of about the solar system. Well, at this point, the sun would be so massive, that all light would fall back towards the center. No infinite density needed! That clears that up.
P.S. That gravastar is probably the star I just described. But that's just my guess.