yes, the hydramatic is what the era automatic is called
- the original floor shift trans is the sm420 4 speed - has a useless granny low gear
- the original 3 speed is a column shift, requiring different linkage and column components than the automatic
either IS a pretty involved swap, won't be cheap and you'll be searching for awhile to find all the necessary parts
- the bellhousing with mounts [and crossmember] was used for the standard trans, a different bellhousing was used for the auto and mounts the back of the engine on a trans crossmember, just a different setup
- gearing for "practical driving" has nothing to do with the trans, any one will have the same 1:1 high gear, standard or auto unless you get an overdrive trans, which is a similar but different can of worms, otherwise the rear gearing is the only way to get higher road speed at lower RPM
- yes, the hydramatic will be worth serious money
you might find the 55-59 assembly manual very useful in figuring out your choices
also, if you were to go to a 'modern' trans, like the T5 [OD], all the same changes to the driveline would be needed, + you'd have to find a new floor pan ["high hump" for 4 speed] and rig your own shift setup that would clear the seat

Bill