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 SingTelÂ’s 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.