I understand that traditionally, certain "moods" are associated with certain keys. I further understand that this is mainly due to the way keyboards were tuned during the classical era, but that now, keyboards are "equally tuned" so that the only difference between major keys is the pitch. (same with minor keys). That being understood, are we missing something in the modern performances of classical pieces that were originally written, and I assume, meant to be performed in their "unequally tuned" keys?
The "missing" in "Are we missing something?" implies a value judgment, so instead I'll just address some of the key differences between equal temperament and historical temperaments.
At least these things would be different:
Older tunings, meantone or tempered, generally had a set of keys where the important major thirds were just, or at least closer to it. This affects the degree of consonance of chords (which are stacked 3rds.)
The tempered tunings generally have different sets of 5th intervals, e.g. the Werckmeister tunings, and thus, as an example, the root chords for different keys have different degrees of consonance.
That the different historical tunings result in different sounds is indisputable. For careful listeners this difference is audible. It's arguable (and possibly subjective) about how these sonic differences affect the emotional/aesthetic impact of music for modern audiences.
This video has a nice demonstration of different tunings, albeit with some unfortunate audio glitches.
Some of the "mood" is cultural, e.g. certain Maqam are said to evoke feelings of a distant desert, which might be tricky if the audience has never even seen such, or for one who does not know that minor keys indicate sadness because their musical culture is based on rhythm, leaving classical works to sound much the same to them.
I don't think so, as instruments were tuned to sound good in one particular key. True, they didn't sound good in another key, due to the tuning vagaries, but in one key, it all worked. So, with 12et, now, we could play a piece in any key we wanted, and it would sound authentic, but what difference would that make?