That is referred to as an instrumentation amp topology, you can find lots of information searching for that term.
Are there any great compromises with this kind of topology
Not really.
With discrete implementations (i.e. using multiple op-amps to build the same circuit yourself, not integrated onto a single device) you have some flexibility to choose higher current op-amps and lower the feedback resistor values. The minimum noise level is mostly determined by the resistor values around the differential amp (the third op-amp with four resistors around it), which sets your signal-to-noise ratio at low gains.
You can also use a full-differential output stage, or connect two diff-amps in parallel but opposite polarity connection so that you get a differential output instead of single ended.
is there some interaction between the feedback with the two op amps that has a positive effect on CMR?
Yes, since the feedback only references the other input, and not ground/circuit reference, any common mode signal (same on both inputs) is then the same at the feedback point, so no current flows in the gain setting resistor, meaning that gain is always unity for common mode signals. At unity gain (i.e. the feedback connection is open between the two sides) you just get the benefit of buffering, i.e. higher common mode impedance if you have relatively high value input bias resistors, all the CMRR comes from the output diff amp stage. As you add gain the diff mode signal is amplified, but the common mode signal is not, so you get a multiplied factor of the gain on top of the output stage CMRR, basically CMRR increases with gain.
