We routinely taxi both single-engine taxi-out/in. This obviously depending on taxiway state (not a good idea when slippery or need for periodic runup etc.) It also depends on how long to taxi / unfamiliar airport, complex taxiing etc. In other words, use your head as to whether it makes tactical sense.
1, We have SOP either to perform start of engine no 2 by heart, or by support items on the normal checklist or in the eQRH. How to is the individual pilots choice.
2, PF focuses on taxi, PF determines when to start no 2 and says "start engine no2". PM (speaking out loud) Yellow elec pump off /APU bleed on/ engine mode select ignition / when bleed psi sufficient: no2 master lever on. When engine no 2 started and stable, PM turns off APU bleed+ APU, then mode select normal (eng.2 antiice if needed) and calls out loud: "engine start procedure completed". We have a hard rule that the after start checklist MUST be completed before initiating taxi, either 1- or 2-engine taxi (reason being after start checklist includes killer items such as clear signal etc form ground staff/ tug removed etc).
3, Obviously, PF is focusing on the AVIATE part (maneuvering the aircraft safely), but at the same time, follows along with what PM is verbalising.
Training Captain with SAS (Northern Europe)