Extending or Modifying Wireless sensor for Tstat
yes you can extend the antenna, both transmitting and receiving. Its a very complicated subject, but it can be done. Look up Helictical antennas. I would also suggest that you look at the radiation pattern for this type antenna. Short of adding more antenna, you might want to try and move the outdoor sensor around. Try to visualize the path the wireless has to travel. If you can move the sensor to a location that has fewer walls etc between the recieiver and transmitter, that would be better. Also try moving the sensor from a vertical to horizontal plane. that might help also.
To answer your question about powering the sensor with 110volts, yes you can do it, but you would have to interface a 3volt transfomer to the sensor. This can be done, but be aware of national electical code, fire dangers and insurance regulations reqarding this, if something bad happens.
now, it if was me,,, I would hack out the thyristor that does the actual temp sensing and extend that outside, keeping the rest of the sensor inside and fairly close to the receiver.. I would also take a 3 volt wall transformer and solder it on to the battery connections and power it from a close by indoor outlet, but thats just me.