I see you made your choice - Good Luck !
Will the Deerstalker bases fit the old-style carbine? - Yes - the "Deerstalker" was the early model name for the .44 Auto, soon dropped due to a complaint by Ithaca Gun Co over it's copywrited "Deerslayer" trademark. All tube-fed .44 autos use the same, 2-piece base system.
How did the Ruger factory rings work? - There were no factory rings supplied at that time. Ruger only began to supply rings years later, when other new rifles were introduced - like the #1 & M-77.
Was it a one-piece base and rings like you see on T/C Encores? - Two piece, Weaver-syle (#68 Front/#47 Rear) bases were available. There used to be both high & low base sets available (different base numbers), but I see that now only one height is made.