Depends on how you want it to switch, but you can wire it from the lighting circuit, or from the ring main if you use a fused spur. 1mm cable will be more than sufficient for the fan.
Yes. Assuming the fan power rating is < 120W, or thereabouts,and the lighting circuit isn't already overloaded, (normal practice is to wire no more than 10 lamps per circuit, assuming 100 Watts per lamp) then you can power it from the lighting circuit as you described. There are regulations (IEE 16th Ed.) about the distance the switch has to be from the shower/fan. From memory, it has to be 1M out of reach when you're in the shower, or something like that.
In relation to switches, inside the bathroom have to be of the string type pull switch, normal switches would have to be situated outside (like light switches)