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There do not appear to be a definite number of stages to launch a craft. Amongst others entries under the 'launch', 'spacecraft-development' tags mention as many as 4 stages. Engines and fuel vary too.
Are payload and impulse the sole criteria to decide the launch engines, and each stage engine? What factors are relevant to this decision?
A rocket has different needs at different times.
At liftoff, there is a need for a level of thrust that exceeds the weight of the vehicle, or it ain't going anywhere. Solids are very good at high thrust, but usually have a kind of sucky $I_{sp}$ (specific impulse). For example, the SRBs from the shuttle produce 2.8 million lbs of thrust each which is an awe inspiring amount of thrust, but $I_{sp}$ is 269 seconds, really lousy. In contrast the F-1 engine (the largest liquid fueled single chamber (Yes RD-170 I am ignoring you!) ever flown) is only 1.5 million lbs thrust, but an $I_{sp}$ of 263? (What? I thought it was in the 300's.) Regardless the point is high thrust is key for a first stage.
Once out of the atmosphere, performance changes, so using a single rocket from ground level to orbit has its own penalties. (Thus the aerospike, where air pressure (and then lack thereof) acts as a nozzle extension for performance reasons).
Also, once you are out of the atmosphere and working towards orbit, you want the highest $I_{sp}$ at that point. Nozzle design is different for in atmosphere vs out of atmosphere. Usually want higher expansion ratio as the ambient pressure gets lower.
Additionally, denser propellants mean less tankage, which reduces size of the stage. Hydrogen (the fluffiest of them all) is the least dense of the common propellants and thus needs huge tanks in comparison to RP-1 which is pretty dense and can get by with smaller tanks.
Aside from the delta V requirements, there are a number of other things that are used to decide how to make a rocket. A few of the key things:
As far as which engine to use, essentially you want to make a good engine that will work with your desired fuel, and at your desired attitude, be easy to make and test, and meet your weight requirements. More than that is probably beyond the scope of this answer.
No, but ISP and thrust are two of the most important.
Also considered are environmental concerns, fuel handling requirements, drive mass, fuel mass, and price of drives, fuel, and supporting structure. A few types of drive, namely ORION and NTR, are not used despite fairly high thrust and ISP, for reasons of politics, namely, extant treaties.
The ESA won't use certain types of power system (fission and/or radiothermal generator) due to the environmental risks in case of launch failure; this also makes the needed drives for maneuver more restricted, since ion drives require rather high power that cannot be provided in the far reaches of the solar system.
