In a single-phase transformer input, one wire getting entry as live, another as neutral. But what would be in case of its output? In output, which-wire would be phase, and which-one would be neutral?
Is it a possibility that both the output would be "live" (of opposite type) (according to the drawing)? or one will act as phase and the other neutral?
The stepdown transformers I buy from market, for small gadgets, (are not like those diagrams (as I drawn)); contain a center-tap. I guess, the center tap is used to resemble the neutral wire. is it so? or there is some-other cause behind using that center tap? does same mechanism (center taps) used in high-voltage transformers also?
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.
Figure 1. Floating output. Figure 2. Neutralised output. Figure 3. Centre-tap grounded. Figure 4. Outer tap grounded.
Neutral (at least in the USA) is connected to ground (at one point, the first panel in the building). Therefore, as shown, neither output wire is neutral. It is closer to your option 2, both live, though both are isolated from ground.
If you don't need isolation, the proper way to use this would be to connect one wire to earth and consider that one neutral.
As far a center tapped transformer, again everything is isolated from ground so phase and neutral don't really apply. Again you could earth the center tap to make it equivalent to US split-leg 240V where both legs are hot.
As for high-to-consumer voltage transformers, outputting 120/240 volts, then normally a CT transformer is used to produce the split-leg 120/240V. The input is typically two phase wires from a 3-phase distribution.