Since, at some point, every such explanation will have to get to first principles–i.e. axiomatic, logically unjustifed statements–then this requirement is putting the us at a bit of a disadvantage here.
Now here’s my conception of truth, which ought to answer both questions:
I consider it to be analytically true that I exist and perceive, and that I do so, at some very minimal level, independent of anything else. I can’t prove this to be true; I hold it axiomatically.
Since there is an I that perceives, there is an I that has a certain impact on the perception of the things that it perceives. (That is to say that I am not a blank slate; the things that I perceive are contingent at least to a minimal extent on me, since I am doing the perceiving.)
Since the perceiving thing has an impact on the things it perceives, its perception has a very uncertain connection to any posited reality. Reality may be out there, or it may not. It may have certain characteristics that I perceive it to have, or it may not.
In order to escape from this uncertainty–something that would be paralysis if you were to worry about it all the time–my conception of truth (and, therefore, of existence and of possibility) is as follows: I say that something exists if there is an overwhelming probability that it does. I consider this overwhelming probability to be the case on the basis of my perception, and of my experience of perception (memory). For instance, I consider it to be overwhelmingly probable that I am typing at my keyboard at this very instant. It is so probable that I don’t even bother, the vast majority of the time, to worry about it.
Anything is literally possible, but some things are probably not going to happen–thus the dividing line between possibility and existence. It is entirely possible that the computer monitor in front of me doesn’t actually exist. It is overwhelmingly probable that it doesn’t.
Herein lies science, which doesn’t make statements about truth, but rather about overwhelming probability.