I am in the UK and so my comments may not (and probably will not) be applicable in other parts of the world.
1. Only if the pilot remains in the same airline. Your position on a seniorty list will determine the number of stripes on your shoulder.
But we have many First Officers on our 757/767 aircraft who have been Captain of a Dash 8 / Fokker 50, even Embrear 145 in the past, with different airlines. That does not qualify them for Captaincy on a 757. In fact many airlines in the UK do not accept direct-entry Captains and so only promote exisiting pilots to 4-bars. Monarch Airlines is an example. If you moved to MON therefore, you would become a First Officer regardless of your present rank or aircraft type.
2. My airline only operates one fleet type (757/767) so this does not apply as all pilots are dual-rated. In BA I think after 5 years or so 737 service a pilot may bid for a fleet change (i.e. go onto longhaul on the 772/744) or opt to stay on the shorthaul 737 fleet and get a promotion to Captain. Financially, there is little difference, but the lifestyle is completely different. Long-term prospects are also better with the widebodies (financially).
3. Not necessarily, in the UK at least. Although generally speaking yes. In the past BA have taken several cadet pilots (sponsored with no airline experience) straight onto the 757 after training (therefore bypassing the 737 and A320 fleets in aircraft size). It all depends where they need pilots, but they will try to aim for the smaller types.
Hope this was of some interest.
I used to love the smell of Jet-A in the morning...