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I've used many 'Tap-Con' (sp?) masonry screws in bricks, concrete blocks, cinder blocks, mortar, and solid concrete and have never had a problem splitting any of the media. I think the key is to use the correct size drill and you'll be OK.

That flue is probably killing you on heating bills moreso than on the AC bills.

Whether or not to seal it off permanently or, as LJV says, make it more or less removable, only you can decide what works best for you and possible resale value.Regardless which approach, you really need to make sure that chimney does not leak air, or varmints getting in, or water dropping down it once the bottom is sealed off. A permanent cap on top will prevent that. I would also think that a good cleaning first will prevent future odor in the house.
Tap-Con screws are usually safe and effective in most masonry, although some soft shale bricks may not grip as well as you want. Make sure that you use lots of appropriate construction adhesive behind the plywood for additional holding power--you will also need fewer ugly screws. I suggest countersinking the screws and filling the holes for a nicer looking finish.
As to the cleanout, try to remove it entirely and brick up the opening outside. Otherwise, it will rust away and critters will move in, as well as leaking HVAC and water.
Fill the firebox with as much insulation as you can get in [don't pack it, just lay it in loosely to get the maximum R-value from it]. It might be the warmest wall in the house, based on the age of it!
Feel free to e-mail if you have any concerns.
Scottw

This might be a much easier solution, and it would be easily reversible if you later wanted to reopen the flue for a woodstove or similar unit:

http://www.chimneyballoon.us/chimneyballoon.html" target="_blank