May a Chazzan decide on his own to do a Hechi Kedushah, or are there rules for when it is appropriate/mandated?
The Shulchan Aruch (OC 232:1) recommends doing this when השעה דחוקה=the time is pressing, which the Rama (OC 124:2) defines as when the congregation is afraid that if they do the full repetition they won't finish by the end of the allotted time for that prayer. (See Biur Halacha OC 124 sv SheYa'avor who debates if this is Chatzot or Sof Zman Tefillah in regards to Shacharit.)
The Mishna Berurah there (sk 6) points out that if it is not a pressing time one should not do this, as the proper form of the enactment of Chazal is to have the repetition come after the silent prayer.
The Aruch HaShulchan (OC 232:7) suggests that, at a very small minyan where it is doubtful that 9 people will answer Amen (as they tend to talk, etc.), one should always do a Heichi Kedusha lest the leader's blessings during his repetition be levatala.
I read in The Making of a Gadol that (according to R' Yaakov Kaminetzky) in Kenesses Yisroel in Slabodka during the week they did a hoiche kedusha for mincha because the institution of chazaras hashatz was for a beis hakenneses where baal habatim davened because of the possible presence of the ignorant, not for a beis medrash of baalei torah. Friday mincha, though, they did a full chazaras hashatz because many baalei batim came for shabbos davening. (I guess that made it into a shul?)