I am looking at the Mitsubishi Dual Zone ductless min-split unit. The 2 wall units will share the same east facing wall and for ease of installation and aesthetics, I was considering doing a dual zone with a single outside unit.

1 - What are the advantages or disadvantages to having two units share a single outside compressor?
2 - Do both indoor wall units have to turn on and off at the same time?
3 - Which setup is more energy efficient; single or dual zone?
4 - Does the outdoor unit constantly stay on or does it just kick on when needed?

Mitsubishi has changed the warranty on these units to 5 years parts/7 years compressor. Plus the heat pump units are good down to -15F below zero.

We install them and I have one in house.

They are one of the most efficient air-to-air heat pumps available. The compressor modulated to the output needed making the unit more efficient in operation since the unit will operate in a wider range of output. You put the terminal units where the cooling loads are.

It is very difficult to condition room without moving air in those rooms, especially when cooling, thus multi-head non-ducted systems.

The outdoor unit (condenser) will run when there is a call for heating or cooling. Mine runs most of the time, but you can't hear it operate or notice the fuel bill for that matter.

Mitsubishi mini-spits are simply brilliant.

Two independent compressors gives you redundancy (ie, some output if a compressor fails). I'd look up the efficiency specs - a single larger compressor should have some cost advantages and possibly some small efficiency advantages.

The interior heads of multi-splits operate independently, but must share mode- you can't have one head cooling while the other is heating.

Both the outdoor & indoor units modulate blower speed, and vary the speed with sensed load. The compressor modulates continuously with load based on feedback information coming back from the heads. When there's no load, the compressors & blowers stop. The loads are continuously sensed via input air temperature(s) at the interior head(s). It's the fact that they are always loping along at part-load that gives them their favorable efficiency, since the coils are sized for the MAXIMUM.

In general separate units have slightly higher SEER and HSPF ratings but in practice that's not always meaningful. The most important factors to get right are to size the heads correctly for the zone loads, and the compressor to the maximum load. Most manufacturers will have smaller heads available for multi-zoned units than you can use with even the smallest single. The higher performance ratings for the singles is a matter of optimized algorithms & compressor speeds for serving a single master. When calls are coming in from two heads separately the outdoor unit often plays a hunting game, ramping up, then down slightly, since the loads sensed at the heads really aren't going to be fully sychronized.

Cost, and lower standby losses usually favor the multi-split solutions, but not always. In cold/very-cold climates standby losses can be an issue, since each outdoor unit still uses power at idle when it's cold out to keep the compressor and it's lubricants at a non-destructive startup temperature. It's not a huge amount of power, but with multiple oversized units running low duty cycles it can add up.