Measuring transceiver output power, e.g. with a SA is the most simple method to verify the matching network, load pulling (varying the complex load impedance until maximum output power is achieved) the next step. In case of a differential output, it can't compensate for a balun phase error, however.
Q1. Depends on. PA impedance measurement during send phase is possible with restrictions. Presumed that the VNA can handle the transceiver output power, it need to be triggered to send phase. Impedance can be measurement with some distance to carrier only.
Q2-Q3 are related to general VNA practice. Ports can be also switched by reconnecting cables, you don't necessarily need a mechanical or automated switch. Single ended to differential S-parameter transformation has been repeatedly discussed at edaboard. With most standard VNAs the calculation has to be performed by an externaltool, e.g. a spreadsheet calculator.
Q4 is a matter of RF chip design. Differential output is usually better for low voltage, also promises better attenuation of harmonics and spurs by balun common mode supression.