1. Mental calculation between current and next charted altitude using a 3-degree slope with corrections for wind against the FMS calculated crossing altitudes. Works like a charm.
2. No other methods, but I verify every few thousand feet, maybe a little more frequently in mountainous terrain.
3. VNAV is used during all descents except for immediate ATC requests. We're a RNP-AR heavy airline, so we don't routinely have to change AP modes during descent or approach - we're generally coupled from TOD to landing. When there is no STAR, distance-from-airport gates are programmed (e.g. 3,000'+field elevation at 10 miles back of airport or MSA 25 back, depending on terrain and conditions).
4. I only use speedbrakes or the props (when flying turbo-props) when ATC required a help with spacing. Otherwise, I'd manage the energy so as to avoid putting out the brakes.
5. We never modify an approach in the FMS for ATC requests because it corrupts our database. SID and STAR requests will get programmed if there is time, but mostly ATC requests are handled through the FGCP.
6. A few times while doing line indoc and maybe once or twice during line ops. It happens, but the easiest thing to do is not to get in that position in the first place. An extra minute in the downwind or slowing an extra few miles sooner makes all the difference. Tell ATC what you need and they're usually accomodating.
7. A few times the pretty pink lines disappear, and we've gone around, but always when in IMC. If the autopilot stops working, no big deal - that's what we all got into this for in the first place, flying airplanes. When the flight director stops working and we're in VMC, the automatics come off, and again we kick it old school. I don't know why you'd go around on an otherwise stable approach with the airport in sight just because some information on a screen you're probably not supposed to be looking at anymore gave up the ghost.