Here's how I would proceed:

1. Relying on the Hart & Cooley documents which we have exchanged, determine
the manufacturers specific structural requirements for our chimney.

2. Decide what portions of the required work, if any, you wish to try to do
yourself, and which you wish to consider referring to a sub-contractor. 

3. Ask the plasterer for the names of
a) a carpenter/roofer and/or
b) heating and chimney contractor
who would climb on the roof to inspect and measure the dimensions
of the existing chimney/rain cap to determine what changes if any
require to me made to meet Hart & Cooley guidelines, 
(and incidentally and unrelated, place cap on existing 3" PVC vent),

4. Decide whether to offset the descending chimney pipe or reposition
the existing 2x6 ceiling joist of the 2nd floor closet.

5. Project the extension of the chimney pipe through the rough flooring
of the second and first floor closets, if possible, avoiding floor joists,
so as to minimize the number of requires offsets.

6. Ascertain if possible from any or all of the workmen consulted what
further permission and/or inspection of the metal chimney will be required,
(without mentioning the circumstance that no chimney is included on the
approved plans.) 

7. Decide whether to proceed with the installation of the chimney with
or without formal approval of the building inspector.

8. Purchase the additional hardware, including pipe which will be needed
from a source to which it can be promptly returned if it does not fit.

9. Install/complete the installation of the chimney into the basement.

10. Build, or have a carpenter build the required inclosures of the
chimney pipe in the space between the second floor closet and the roof,
and in the two closets through which the pipe descends inton the basement.

11. Consider instead of installing a metal chimney, the possible use of 
the existing 3" PVC vent as a chimney for a gas-fired furnace, 
and abandoning the contemplated metal chimney.