For what I know after reading the MIO FAQ:
1) In order to enjoy the unlimited free local internet call, You
need a VoIP-enabled mobile phone and WiFi Internet connection to
make Internet calls. Currently, only Nokia handsets namely, E60,
E61 and N80 IE (Internet Edition) support mio mobile service. The
firmware of Nokia E61 and E60 handsets will need to be upgraded to
support the Service.
WiFi internet connection means your home wireless connection or the
hotspots located in Singapore. Even if your neighbour's HP/PC get
access to your MIO box, it will not cause any interference to you
except your neighbour get to use free internet.
As for will your neighbour be allow to access and use your wireless
internet connection, the MIO box also act as a router, that means
use can set WEP/WPA encryption security to only allow those who
have the key to access your wireless network.
For your info, MIO mobile do not give unlimited free local call
unless you are making call using VoIP aka Internet calls.
Traditional method calling follows the plan you are subscribing
(e.g. 100 min free local call or so found in the plan).
2) About your question on the unavailable of MIO vioice. As you
know MIO voice is a SingTels new digital voice service offered
which allows customers to make unlimited local calls at home. That
means it uses internet to make calls, which also means your home
phone will be connected to MIO box. So what will happen when there
a power failure? Yes the MIO box will crease to work, therefore
cause MIO voice to be unavailable (this is what singtel means by
MIO box is unavailable). In the other sense, no power = MIO box
stop working = no MIO voice, you will have to use your HP or fixed
line service (traditional method) to call.
Yes, you have to leave MIO box powered on all the time. mio box is
a low-power device, so it consumes very minimal electricity.
Typical power consumption is below 13 watts, which will cost less
than $2 per month (estimation calculated from SP Services).
Below is the diagram how phone and MIO box is connected.

Let me tell you something, for more info on MIO plan/box/voice or
mobile, go to their respective FAQ. It helps alot on clearing your
doubts.
*All the infomations here is taken from the FAQ
of MIO website.