Well, it is in general hotter (the scale isn't ordered by temperature but presence and absence of particular spectral lines, but those criteria coincide to some extend in many cases). The relation to mass holds only on the main sequence. Red (super-) giants are usually M class but can be much more massive than K, G or A dwarves, for Antares I find M class and at least 15 solar masses. They just puff up, shedding only little mass, and cool down on the visible surface, which makes them M. White dwarves on the other hand are usually hot and to the left on the spectral scale but not very massive, for Sirius B I find B class at solar mass.

At this HRD the former record holder in our milky way is Eta Car, the "X" at the upper left at (B0,-10), the extended (puffed up and rather thin) biggies are at the top right, VV Cep being one of the largest at all.

I haven't read everything about the NGC3603 issue, but it seems to me they just labelled the items sequentially starting with A for the brightest (maybe appending a number in case of evidence of it being a multiple itself), that's common practice for multiples. It's got nothing to do with spectral class. Some prominent examples:

Alpha Cen: A is G class, B is K class
Procyon (Alpha CMi): A is F class, B is a white dwarf (looks like A class)
Keid (40 Eri): A is K class, B is a white dwarf, C is M class