In general, the proper setting is going to be 75 ohms. Here's why:
When used with a PC source and mini-coax breakout cable, the 75ohm termination matches the characteristics of the cable and back-termination on the PC. This will reduce ghosting and similar due to transmission line effects and yield proper signal levels. The same goes for arcade boards with similar outputs (some boards with HD15 outputs, especially the ones with support for 31kHz mode).
When used with conventional JAMMA arcade boards, the 75ohm input will load down the board's output and dim it. This sounds bad, but arcade boards like this conventionally output much higher signal levels than the monitor expects (~3.5V vs. 0.7V). Many arcade boards were designed deliberately do have proper signal levels for both arcade monitors with Hi-Z inputs as well as PC type monitors with 75 ohm inputs due to careful component selection. If the video is REALLY dim (this will happen with some boards, such as MK2, depending on how they were built), go ahead and try the Hi-Z setting.
You can use this as a crude brightness/contrast switch, but you shouldn't. You probably won't break anything if you have it set wrong, though. The increased intensity may cause premature tube wear (burn/dimming), at the extreme end.
If you can't get the monitor "bright enough" on the 75ohm setting when being driven with a PC, you may need to adjust the color gains/cutoffs in service mode. Also consider that it may just not actually be intended to be that bright (see the burning/long-term dimming comment above). Many people have been conditioned by years of "brightness wars" on TVs as well as modern contrast tricks and excessively bright backlights on LCD/plasma to expect super bright and contrasty displays with overly saturated color (this will combined to look "vibrant"). I guess it's personal preference, but the "proper" (photorealistic looking) setting is generally much "flatter" and overall "dimmer". I tend to set my monitors to this setting, with the side effect of making them last longer, but again it's your monitor in the end, and you should make it look the way you want it to. Many people who have me adjust their monitors for them initially comment that it's "too dim" but quickly decide it actually looks better.
The setting could be used for pass-through video, as you note, as well as for using passive splitters. This will ruin the beneficial termination effect, but it at least keeps things from getting excessively dim when this is done which was sometimes a worthwhile trade-off in a commercial environment where these monitors often needed to "just work" on a field setup.
The switch under the sync input is probably a similar function. It usually won't matter for TTL sync as used by PCs and arcade games. Some professional video applications had sync that was different electrically and either needed or benefited from (on long lines) the 75ohm termination.
As a note, some arcade monitors have a similar switch. On them, switching to the "Hi-Z" setting not only changes the input termination but usually also adjusts things to expect the higher signal levels, so on them you want the Hi-Z setting when used with JAMMA boards and the 75ohm setting for use with a PC. Confusing, I know.