From HDMI 1.4 upwards HDMI connection can be made in a home cinema set-up.
For mixing when you go from Multi track to 5.1 it used to go from DTRS with either 48khz or the much rarer DSD-1 with 8 tracks synced up to how many boxes you wanted (8 stacked made for the Eagles recording) into an analogue mixer. Into DASH or maybe analog 2 track tape. Onto a digital master recorder. The first chain was allso replaced at some studios with a full analogue setup onto 24 tracks/cascaded 48 tracks analogue tape. Then digitized. Mixed digital then into end media's. And some even will mix analogue 5.1. Where only the last stage digitizing will start.
With Protools it would be analogue preamps into A/D into Protools into Multi track project with controller into 6 stems. And then mastered. After mastering encoded into DVD or digital streaming format.