I've just realised that to have the system working all the time, I will need to have, besides the 2 USB cables for serial communication, 2 power supplies (for when the computer is disconnected). There's anyway that I could use only one PS, say, for the Seeeduino R3, but keeping the features discussed above? (2 USB cables and I2C connection between both and the RTC)
If you supply one Arduino, you can just connect the other with GND and 5V pins (on both MCs) and you have power for both. This works if the second Arduino doesn't pull too much power (the RTC is absolutely no problem).
If not, can the Leonardo "login the network" as a slave at anytime, i.e., when the computer starts and be recognized?
To wich network? In the previous posts you haven't had any network equipment involved. And what does "slave" mean in this context?
I wouldn't need one more Arduino unless the EasyVR Shield were compatible with Leonardo.
Unfortunately there doesn't seem to be any documentation about that shield. The library is fully compatible with the Leonardo, my guess is, the shield is too. At least I couldn't find any specification that would rule against that. Do you have a schematic of the shield (I couldn't find one)? The EasyVR module is completely compatible with the Arduino Leonardo. If the shield isn't done too badly it should be as well.