Chris...I can make a few suggestions when owning a t-bird and i feel qualified as I have owned 1/2 dozen 56 birds thru the years.
#1..join CTCI .....classic thunderbird club international.
#2..
http://autos.groups.yahoo.com/group/...guid=301108479
paste the above and log on and sign up to this t-bird group...
#3...purchase the 56 Ford repair manual from CTCI
#4...purchase the t-bird illustrations and reference manual from CTCI
#5...purchase the electrical manual from CTCI
as far as a spare,I use a donut mini spare from a 1988 lincoln town car. This saves tearing down the connie kit if you have a flat tire...or carry a full size spare in the trunk.
As far as the heater core...I have found that the first thing to check is the heater control valve on top of the block,this is part # 18495. It looks like a giant mushroom. These are notorious for going bad. Start with this item first.
BE SURE TO SIGN UP FOR THE YAHOO T-BIRD SITE. You can search out any t-bird problem there. These are the most knowledgeable people around for the t-bird...good luck and enjoy your car...John in CT