My wife's 1999 Accord has this situation. When accelerating at a
normal pace and the engine (automatic) shifts into 3rd gear and you
then push your foot down to pass (i.e. getting on the expressway) and
the engine down-shifts into second gear, there is a slight hesitation
with the engine still reving before the car takes off.

I went to a transmission place and they said there was "metal debris in
the system". He said they did not know what kind of metal but they
would have to take apart the transmission to look inside ($500 labor)
plus the cost of parts. Is there any truth to this or is he just going
to not do anything and say something like "well, the parts look
good...cash or credit?"

If there is something wrong will it eventually hurt the car and make it
(maybe) not operable, or can I just not floor it in this situation?

Thanks!!

Eddie G