Imagine a pair of mono-bloc amps built in ONE chassis, and with one power cord-- that's what a dual-mono amp is. A mono-bloc amp has only a single output-- to just one speaker, whereas a dual-mono amp has outputs for two speakers (left and right) and is thus also called a dual-mono stereo amp. Note: many "regular" stereo amps are not dual-mono because the two channels share much circuitry. Theorhetically, dual mono amps should be better (and more expensive) than regular stereo amps.
Dual-mono amps are pretty much by definition also fully balanced as each half of the dual-mono amp has identical but separate circuitry. But, as Swampwalker said, stereo amps can have balanced terminals (XLR) without actually being a fully balanced design at all.
Some fully balanced components don't have RCA outputs, ie BAT (I think), but most fully balanced components have both XLR and RCA outputs and inputs just for convenience (and saleability?). Finally, any stereo component can be fully balanced because they have two channels, ie transports, DACs, pre-amps, as well as amps. If you believe in balanced construction, it's probably best when used on all components in the system. Personally, I have all fully balanced components but prefer RCA ICs-- just my preference. There is no consensus as to which is better-- XLR (balanced) or RCA (single ended), and each has it's proponents. Good Luck. Craig