What limits the use of the H-R diagram to measure distance (main sequence fitting), what distances is it useful for? Is it only possible to measure objects that form around the same time? Is it possible to measure clusters from distant galaxies other than our own?
It depends how precise you need to be. Main sequence fitting assumes that the star(s) in question is(are) on the main sequence. If you have a cluster of stars (at the same age) then defining what is on the main sequence and what isn't becomes much, much easier and of course you have lots of stars with which to beat down the statistical uncertainty.

In fact, for coeval groups of stars what you term "main sequence fitting" is rarely done. The process is the fitting of an "isochrone" (a line that links points in the HR diagram at a single age), so that both the distance, age (also extinction and metallicity) are possible free parameters.

The subtle distinction here is that what is termed "the main sequence" does not really exist in practice, or at least not as a uniquely defined locus in the HR diagram. Star begin their main sequence lives on the "zero age main sequence" (ZAMS) and end their main sequence careers at the "terminal age main sequence" (TAMS), gradually changing their luminosity and temperature as they do so.

Here is a diagram from Martignoni et al. (2014) showing the ZAMS and TAMS for stars of different masses. there is typically a factor 2-3 in luminosity between them (a larger difference at larger masses). That means whether you use the ZAMS, the TAMS, or something in between to determine the distance from a vertical displacement in the HR diagram, you could vary your answer for the distance by $\sqrt{2}$ to $\sqrt{3}$. In other words you need to know the age of a main sequence star before "main-sequence fitting" can give you an accurate distance.

Of course, lower mass stars are longer lived. Anything of say 0.7 solar masses or below has hardly moved from the ZAMS in the age of the Galaxy, so there would be little error in assuming a ZAMS locus. Conversely, the effects of age are far more rapid and therefore far more important on the main sequence at higher masses.

If you were to try to use main-sequence fitting to estimate the distance to individual stars then there are several hazards. For one, it can be nearly impossible to estimate the age of an individual star. Therefore if it has a mass greater than 0.7 solar masses then there will be an uncertainty in its position in the absolute HR diagram that leads to an inevitable uncertainty in estimated distance. Further, the intrinsic position in the HR diagram depends on the star's chemical composition. Such ancillary information might be available, but it might not, in which case that is another source of error. A further source of systematic uncertainty is stellar rotation. Fast rotating stars have extended lifetimes on the main sequence and somewhat different intrinsic positions on the HR diagram; again a source of systematic uncertainty that is especially problematic for high mass stars. Finally, it can be that what you think is an isolated main sequence star is in fact a binary system. A companion can increase the luminosity of the system and make a star appear closer than it actually is (by up to a factor $\sqrt{2}$).