3d printer [need advice]

Yes you always need an extruder no matter how you get the carriage mechanism.

It's not that cheap, do you know somewhere where i can get one for about 20 euros?

A hack saw? Just a few Euros.

Do you think I will have enough precision for what is required?

The couplers are included in the plastic parts (Z axis) - the other two axis use belts.

Does this mean I only have to buy the motors, plastic parts and threaded rods and I have all the mechanic part? Where to get the belts from?

And how to make sure the motors I'll buy will fit with the printer?

In my eduction place i don't have any machines as we're doing maths, physics and chemistry but no engineer work.

Thanks a lot for all your answers!