Back on the aircon topic, split systems are the way to go, just make sure they are of the 'inverter' type that is vastly more efficient having a soft-start feature and also variable running speeds for the compressor rather than the old 'on-off all or nothing' type. I think most split systems are of the inverter type these days anyway.

Panasonic, Fujitsu or Daikin are all good brands here (we have Panasonic) and yes, the interior unit is powered from the exterior unit.

Another plus for split systems is that they are quieter since the compressor is further away.