Necessity of a current mirror circuit - MOSFET/BJT?
The V-I characteristics of the diode-connected transistor and the controlled transistor need to be the same.  You wouldn't get that with a diode and a transistor.
Current mirrors are used in integrated circuits, where transistors are cheap and most other components are expensive.  How would you control Vbe to get the collector current you want?  How about using the cheapest component available to do the job?  That cheapest component is a transistor.
The collector currents are the same by virtue of the fact that the transistors are matched.  Because they are matched, the Vbe generated in Q1 matches the Vbe needed to generate the same collector current in Q2
Q1 is not in saturation.  Saturation happens when the base-collector diode is forward biased enough that current would flow from base collector in the absence of emitter current.  Vbc of Q1 is zero, so the junction there is not forward biased at all.