3. It does seem like an external serial modem is the next best option, although then I have to provide power to both the Arduino and the modem. The power in the building is not entirely reliable. I could run the Arduino off of the telephone line, I think, but I'm not sure how to do that for the modem, which uses conventional 120 volts. I suppose I could take the modem apart and try to supply whatever it needs internally.
Actually, the modem is likely something like 5 - 9v DC. The wall mounted transformer steps it from 120v AC to whatever it is that it uses. I think I've seen external modems that input ac at 9-12V but I think most are DC.
As for the building power, you could run everything off of a battery and charge the battery when you do have power.
4. In order not to need a dedicated computer elsewhere that the Arduino is calling, I think it might be better to have the Arduino accumulate temperature data in its own memory and then I will call it when I want to know what's going on. I'll have to figure out how much memory is available for data storage. And this would require the modem to be on all the time so that it could listen for my call.
Well, the arduino has 1K ram, 16K flash, and 512 bytes EEPROM. If you store temperature as a signed byte (plus or minus 127) you could store 512 readings in EEPROM or maybe 700 - 900 in ram depending on how much ram you use. You could even use progmem and store things in the flash but it's more complicated. Then you could conceivably store upwards of 12k temperature bytes.
I still say you are better off calling when there is a problem and letting everything stay low power until then. But that would require you have something which it can call. A pager or cellphone perhaps.
Anyway, there has been a lot of ideas posted here. Hopefully they all lead you toward a good solution. Best of luck and don't hesitate to keep chatting until you've got something great!