In the "getting-to-know-you stuff", the hyphenated part is a 'modifying phrase'. It basically fulfills the role of an adjective. Outwardly it reminds a 'compound adjective', but does not have the typical characteristics of an adjective.
The key thing that makes it work like an adjective here is that it is in the attributive position relative to the noun "stuff". That is, it stands before "stuff", and describes it, like an adjective would.
With the hyphens, it is easier to understand that what you have before your eyes works as an adjective:
What kind of stuff? The getting-to-know-you stuff.
Quirk et al. describe the use of the hyphen in Unit III.4 of their "Comprehensive Grammar" (page 1613). They basically list the cases in which it is used, and point out that it is used "to avoid misinterpretation".
So, technically your example could be a 'modifying phrase', listed as no. 4 (out of 9):
(4) other modifying phrases and modifying clauses (which are generally written open when not modifying), eg: on-the-spot (investigation), face-to-face (meeting), ten-item (test), right-to-life (movement), up-to-date (news), take-it-or-leave-it (attitude), do-it-yourself (job).
The English poet J.M. Hopkins was especially fond of coining compound adjectives:
I caught this morning morning’s minion, king-
dom of daylight’s dauphin, dapple-dawn-drawn Falcon, in his riding
Of the rolling level underneath him steady air...