If you give a certain roll input on the yoke, do you always get a certain amount of spoiler and aileron deflection, regardless of speed or altitude?
No. With conventional controls, some ailerons are locked out at high speed. However, the throw on the remaining surfaces will be the same.
With fly-by-wire, the computer figures it out and does what it wants.
The books say that ailerons are not useful at high mach numbers, because of wing twisting e. a. But do ailerons still work at these speeds in the same way as on low speeds?
Normally yes. The word "high" is relative, dependent on aileron position and wing stiffness.
However, "aileron reversal" may happen at speeds above Mmo.
[quoteIt is well known that yaw dampers are used to avoid dutch roll, but do they also give turn coordination?
Because as far as I know, the rudder of a jet is not used in normal flight with the gear up, in normal cases. Is it perhaps because roll spoilers also give a yawing moment?[/quote]
Depends on the installation... Turn coordination may be a function of flap position, airspeed, or what the computer decides. Spoilers do produce yaw in the proper direction, unlike the adverse yaw caused by ailerons. Airliners are usually rigged such that manual rudder input is not needed in normal cruise.