My question is, do I need to bring the stove pipe all the way up the
chimney to the roof (house is two stories and an attic)? The chimney is
already capped with a raised flagstone so that rain water does not come
down the chimney. Is there an advantage to going to the top, or am I ok
just having the pipe exhaust into the big old chimney? I'm mainly asking
because the old chimney is the original stone chimney and I think it
might be safer to use pipe to the top, but maybe there are other
considerations?

Modern wood stove flues are usually designed for higher heat than your
existing chimney, and are ideally corrosion resistant double walled
stainless steel. It may be best to run the pipe all the way up. Re whether
to seal around the base of the pipe, I don't know, but a good newsgroup for
this would be alt.energy.homepower
With a hearth as large as yours, a stand-alone stove might work better than
a woodstove insert, which, as I recall, generally fit into smaller openings.