Is it possible to have a wood burning stove by my fireplace and vent the smoke through my existing chimney?  Fireplace wasn't built right, never use it.  But, the chimney is in great condition.

Yes if its a masonry you can do a direct connect through a thimbal pass-through you will still need to have a liner though it must be a direct connect from the stove all the way to the termination

Why not install a wood burning insert into the not right built fireplace?  If the chimney is good?  Serious questions...not jerking you around.

This is the correct path.  A wood burning insert will make it extremely efficient.

I had the idea of being able to cook on top of the stove, but yes an insert would work. Ill look into an insert, theres a place in Georgetown that sells them.

Just be careful of creosote build up in the chimney.  Just look it up.

After 45 years of doing this and also owning a stove shop. I can tell you whether  you go insert Or free standing you will need to do a direct connect liner for stove to work efficiently and not to become a creosote factory. We currently charge 249 to sweep a single Chimney and 450 To pull a insert and cleaning if glazed can cost up to $750. Much easier to clean a direct connect.  Trust me on this  my advice is Pay on the front side.  Install a insulated liner on the stove it will make the stove operate more efficiently use less wood and create less creosote. You can also go to the freestanding route this will give you more radiant steel in your room for air to get heat from if you have a large area this may be preferable to an insert. As you pointed out some models you can cook on top of