The only time one really needs to be on autopilot is doing an autoland or in RVSM airspace.
Most hand fly after takeoff up to 18-20k or so, of course that varies widely. There is no company directives(with us) on when the autopilot has to come on.
When descending, weather may be a factor, plus you load up the other, non-flying pilot some with working the MCP panel and watching things more closely.
There is no average on when the A/P comes off. With a visual approach, maybe 10 miles & 3000', again varies widely. When something extra is planned, the flying pilot often mentions ahead of time that they would like to do extended hand flying.
I've had friends that have flown in the Mideast, seemed to say it was a lot more regimented dealing with the A/P.