Can anything exist for an infinite time or outside of time?

Just to answer this question, there is nothing in the current laws of physics to suggest that something can't live arbitrarily long into the future.As for having an arbitrarily long past, well there is nothing in the laws of physics to suggest that can't happen either. It can't happen in our universe because our universe is not infinitely long in the past direction. However this isn't a logical consequence of General Relativity or Quantum Field Theory, it's just a feature of our universe.Of course it could be that a later theory of physics will have our universe's finite past as a logical consequence.

That's also not necessarily the case. For instance in General Relativity it is possible for a particle to live forever in the past direction and at some finite time in the future have its "life" end by falling into a black hole. It sounds bizarre though and of course it's can't happen in our universe, but Quantum Field Theory and General Relativity don't logically forbid it. Yes, nicely spotted. Basically because it's a more accurate way of saying what is going on. At any arbitrary point in the past the object still exists, no matter how far in the past you look. This avoids the confusion usually associated with infinity and is basically the mathematical notion used in General Relativity.