How to calculate current limiter resistors for ULN2803 and Relay?
The ULN2803 is a darlington transistor array, meaning it works basically just like a normal transistor - a current controlled switch.
This switch does not push current through the relay, it is merly closing the circuit "VCC-Relay-ULN-Ground".
You can also see this in the datasheet: the 500mA are an absolute maximum rating, not a characteristic that is simply always fulfilled.
Therefore, you do not need any current limiting resistors on the output side.

If you used a single transistor, you should use a current limiting resistor on its base, but this is handled by the ULN.

Concerning pullup/pulldown resistors: I would recommend to use a ~10k pulldown on the input side. Generally you don't need it, but it can prevent the relay from switching on when the uC is in reset state.