"George's body is composed of matter, right?"

Yes.

"When does that matter cease to exist?"

The matter can still exist, though, when the body does not. If I chop up the body into little bits, and scatter the bits to the wind, the body is gone, but the matter still exists. hence, the body is not identical to the matter that composes it.

When you think about it that way, it seems that George has always and will always exist, no?

No.

George's mind is a construct, an abstract notion. In that sense, did George ever actually exist?

This seems confused to me. On the one hand there is George's mind, and on the other hand there is the idea or the concept of George's mind. They are not the same thing. Both are real, I would say, i.e., both exist. The usual issue with positing the existence of something like a mind is that it is not physical, so a physicalist would either want to say that the mind does not exist (if it's supposed to be a non-physical thing), or that the mind is actually a physical thing (maybe in that case identical with the brain). I don't equate George's mind with George in either case, so regardless of one's stand about the existence of mind's, that George exists is not a problem (I say George is a wholly physical thing--a human animal).