Would this setup work?
I don't know, but here's an approach to find out.
A way you can very roughly estimate this is to calculate the volume of your assembly (i.e., 80mm long, 40mm in diameter). Then find a RC airplane motor that has about that much volume externally. Finally, take the torque that motor will generate, and turn that into the equivalent force you'd get if you put a lever on it that travels a total of 80mm in one turn of the motor.
If that force is within a factor of two either way of your required 150 Newtons, then you have a chance. If not -- look at alternate ways of achieving your ends.
Is it quicker and more effective than a traditional motor driven linear actuator?
Possibly, but you can find that out by doing some math. Find a 40mm diameter motor and get an idea of what torque and acceleration you can get out of it. Then see if you can find a suitable jackscrew that'll turn your available torque into linear motion.
How should I design the Solenoids to get the 15kg force(what wire gauge, how many turns, diameter etc...)?
The predictive-science may is to get a FEMM program and do some designing. If you don't have experience with that and don't want to do the equivalent of a self-taught master's degree in EE, you may want to find a consultant with an FEMM program.
The experimental-science way is to wind some solenoids and experiment. Wind up some solenoids of various dimensions, and see how much force they'll exert on magnets of various dimensions. Be sure to figure out how much current you can put through each one without it overheating. This should give you an idea of whether your idea is even practical.
Note that a nice characteristic of solenoids of this sort is that for a solenoid of given external dimensions (i.e., the inner & outer diameter, and length), the magnetic field it generates is pretty close to a constant times the power it dissipates via resistive heating -- it just works out that way. So if you hit the right dissipation & force numbers, but the voltage is too high or too low, you can just rewind with bigger or smaller wire.
How many solenoids should I have to achieve 80mm stroke in 50-100ms and what length should each solenoid be.
This is part of your previous question. Find out the ideal solenoid dimensions; that'll tell you how many you need.