frame size/shimano?
 

does it matter what your road frame size is and what your shimano combo is? For ex. (totally hypothetical), if I got hand-me-down bike that was too big (say 56cm) with ultegra parts, could I transfer all or some of the ultegra components to a 50cm bike that has 105 parts?

yea, but it might depend on wether you have a triple or double on the 105, most ultegras only come in doubles