Ispahan
Yes there are still 2 month temp bases for SOs. Depends on seniority, the bases available for your fleet, if there are enough people wanting one and a new rule just introduced is you must be legal to work where you plan on doing the temp base. Very much hit and miss.
Em773ER
On the airbus fleet (will be almost the same for all fleets)
Most patterns are 3 days meaning you leave on day one to Melbourne for example, spend one night in Melbourne and then fly back to HKG. Usual layover is about 24 hours.
If flights to a destination are daily or twice daily, three times a day or four times a day every day of the week the pattern will be a 24 hour layover most likely. Auckland at the moment (winter) is 10 flights a week so some patterns are not just a 24 hour layover and are more like 48 hours. Moscow when it starts in July is only three flights a week some some patterns will be 3 nights in the hotel, should be fun.
As there are lots of based pilots you do not normally spend the layover with them as they go home but if flying with HKG based crew yes they are normally on the same pattern but there are exceptions. We use SOs to Dubai and quite often the FO and CN do a shuttle flight to somewhere else in the region and so their patterns are longer, 5 day I think. The SO goes back on a flight, operating usually but sometimes px (as passenger) anywhere from 12-23 hours after arriving.
It is not correct that SOs only work on flights longer than 8 hours. The return leg from Dubai is sometimes 6 and a bit hours, legally an SO is not required but you often operate so they can sell the seat they are required to provide in business class if you are on duty travel as a pax. Also some flights less than 8 hours legally require an SO. One of the Airbuses most hated flights for an SO is px to Bangkok, next evening fly to Karachi and back to Bangkok have a day and a half off in Bangkok then fly to Karachi and back again then sit in business class for the Bangkok to HKG bit. Each leg is under 6 hours. I dont mind it as get a descent amount of time in Bangkok. You do not fly with the same CN and FO on this pattern on the second KHI trip.
Yes SOs are on all of CX aircraft types. Not 100% sure if the 777 SOs are on the 777-200 and 777-300 but are on the ER. If on the Airbus you do about a year on the A340 only and then do a CCQ course and operate both the A340 and A330. I am sure you know they are on the 400 including freighters sometimes.