Two sequences in order, one button push
What you could use for this is a 555 timer chip wired up to be of the monostable configuration. This is activated when a low voltage is present at it's 'trigger' pin so you could arrange use the button in a pull down configuration so that when it is pressed, a 0V (very low signal) will be present and allow the monostable to trigger.

The output time can be worked out using the calculation: 1.1 x R x C
Possible values for this can be found using online calculators to save yourself the trouble: Link to timing calculator.
So this could power your moving man and what not as it will output a high when the button is pressed, and you could use the final falling edge of the output to start your card dispenser. The output could be inverted so you could use a rising edge to trigger something, I would a think a type of latch, but that would hold the output high, but if this isn't a problem, ie. cards wouldn't come raining out, then this would work well. There is plenty of material to be found online about  latches, j-type and d-type flip flops, and the output of the monostable could be used for you clock/ trigger.