Caveat: I don't have the kit, or the book.
- Why do we need the 10k Ohm resistor as a pull-down resistor after the push button and what would happen if we left it out?
If you left out the resistor and with the switch open, the input to the processor would float, which means that sometimes it might read as HIGH and others as LOW.
The pull-down ensures it reads LOW when the switch is open.
A cheaper option is to invert the switch logic, and use the processor's free built-in pullup resistors, which ensure the input reads HIGH when the switch is open.
- Why is it necessary to define the digital pins 3-5 as inputs?
I don't have the code, but I'd expect them to be outputs.
wouldn't it also be possible to just use the Arduino's 5V to supply the LEDs with current? (Maybe to be able to make them blink individually?)
The 5V is on all the time, so the LEDs would be too.
The LEDs and the resistors are in series. Why don't we need to put the resistors before the LEDs and why apparently does the sequence in a series circuit not matter?
It doesn't matter here where you put the components, the important thing is that they are in series.
I'd like to find out what the smallest resistor value would be for this circuit. How can I calculate the resistors needed value with the formula R = U / I? (U must be between 1,6V and 2,2 V for a red LED, correct me if I am wrong, but what is the value for I?
Until you're happy that you're not going to damege either your LED or your microcontroller, it's safest that you don't experiment.
However, subtract the forward votage of the LED from the supply voltage, and divide by the maximum forward current for the LED, making sure that his value is nowhere near the absolute maximum 40mA for a single I/O pin.
I would also like to know the same for the pull-down resistor: what would be the lowest value for this resistor and how can I calculate it? R = U/I : U should be 5Vs but how can I find out I?
A pullup or pulldown wouldn't normally be less than about 5kohms.