In a single-phase transformer output, which-wire would be phase, and which-one would be neutral?
It's very simple. Neither is neutral until it is "neutralised" by connecting it to ground. 
Until this is done the outputs are floating or isolated with respect to ground.
Yes, polarities will be opposite.
Again, until the centre-tap is grounded all three terminals are floating. When grounding one could ground either the centre tap or one of the outer taps to get various voltages. e.g., a 110 - 0 - 110 transformer might normally be centre-tap grounded to give two 110 V outputs 180° out of phase but could be grounded at one of the outer terminals to give a 0 - 110 - 220 V output.








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Floating output. Figure 2. Neutralised output. Figure 3. Centre-tap grounded. Figure 4. Outer tap grounded.