How can I turn off automatically, say after a fixed number of cycles, a 555 in astable mode?
You can do this with a pair of 555s, or a 556 (just two 555s in a single package), by using one as a monostable to enable the other, acting as an astable, for a period that generates the number of pulses. 
If I were doing it, it's probably cheaper and simpler to program a PIC or similar to do this, there are plenty of 50 cent microcontrollers that will do this - plus you could use another pin or two for up-down adjustment of the pulse number.