I'm pretty sure it is a humidity issue too. Hot and dry. It gets dry enough to deal with static electricity, especially in the winter. In the winter it is hot because of the heat and location (upper level floor). In the summer it is also hot. The air conditioning unit can only be fit in a wall space. The wall space isn't the plant room. Anyways I took most of the begonias outside. We will see how they do. There are still a couple in my apartment. One is in a ziplock bag and the other in a clear plastic bin. Those two seem to be okay so far. But they are also outgrowing their covered space, so what to do when they get too big to fit their container? Now what? --- Re tap water: Most of the time straight from the sink. Sometimes directly watering from the sink or shower. The shower has a detachable head. If I have any water siting around it is because I was either lazy after filling it, or filled it before having to go some place. Watering can is empty most of the time, even after filling. I have wall planters with some vines that take up a lot of water. If I have any water left over from my drinking cups, I use that too. I have about 2 watering cans I use. One regular sized and one large one. Hard water is a huge issue. There is also a some environmental political issue about there being too much nitrates in the cities (or nearby cities) water supply.. The water tastes terrible too. Kind of a fertilizer/ fishy/ algae/ chorine like flavor. Yuck. It comes from the Mississippi River. I finally resorted to using a filtered pitcher to drink from, but I need it more than the plants. I'll see about eventually getting a 2nd one to use for plants. It is just a less priority than staying hydrated myself.

I can sympathize with you concerning white flies. I don't remember ever having them on my begonias but some other plants can be heavily infested - mainly in the mallow family (hardy hibiscus and flowering maple to name two). I think by cutting off most of the leaves and discarding them to get rid of the next generation has helped them tremendously. If you haven't noticed any signs of white fly in 4 weeks, it sounds like you have the problem under control. There is always the chance of some eggs that may be "dormant" seeing how insects survive even the harshest winters.

I hope you didn't bury the rhizomes. This can cause a lot of problems if you bury them (too deep). This looks like ones from Exotic Angels (Lowe's, Walmart) and usually thrive outdoors in summer. Do not start in direct sun but slowly acclimate them to more sun over a couple of weeks so they receive a couple of hours or more. Even in shade they should do okay. I had several of the small starter plants a few years ago. Wonderful summer growers but a bit fussy in winter (mildew, semi-dormancy, etc)

I know a lady who used AV pots for her begonias and claimed they did great for her. I have 4 such pots but only put AVs in them. Anyway, they are worth a try? As for growing begonias in AZ, that is something you will have to figure out with the dry air. I would go only go so far out on a limb and say you will probably come to the conclusion that is easier to grow begonias indoors where you can keep humidity higher than the outdoor air. I've visited AZ twice and know there is a huge difference between Phoenix and Flagstaff. At least in Phoenix I saw huge Saguaro cacti, agaves, and Pride of Barbados bushes growing without a hitch. Flagstaff looked so green in comparison.