20 seconds of smoke at the start is quite normal. I wouldnt worry
about that. Since the damper is actuating funny you would be looking
at a chimney balloon ($50) or a replacement damper ($500 or so
installed) to fix that issue. If the condition of the inside of the
smoke shelf of the chimney is in question, then the whole interior of
the chimney would be in question, so you would really be looking more
at a stainless chimney liner. But that would be up to a sweep that has
a camera to inspect it well.
Or you could just plug it and forget it, since traditional fireplaces
are such a loosing heat energy option anyway. Using them makes your
energy bill go up - not down.