1. If relays are powered from external power supply, should the grounds be connected?
2. Where to put capacitor and why it is needed?
3. May I use arbitrary pins for data/latch/clock?
4. What will be the state of output pins on power on/restart?
I2C port expander to a pair of ULN2803As to your relays should be enough. The relays don't require regulated power and their current needs are in their datasheet. Get a supply that has at least 50% more current then you need.What are you switching and which relays are you planning on using?