What source are you using for your water? It could just be calcium build up from hard water or it could be ...? could you describe where the tree is being kept and what sort of soil you are using, that sort of thing. Might help someone else make a more educated guess than mine.

To directly answer your question on whether you can pick the green fruits from the plant the answer is yes, you can. It comes at a cost though. If the fruits are completely green, they won't ripen off the plant. If there is even a little blushing to them they likely will. There are a number of chemicals that contribute to the overall flavor of the ripe fruit though and removed from the plant, they don't make it to the fruit so there is often some flavor loss when ripening off vine. I am really sorry to hear about your experience, but I would recommend the same thing as Raybo, find a fungicide that is food safe and seems likely to be effective against common foliage pathogens like PM and use them as needed. There are organic and synthetic options, but I don't know what is available to you where you live. It sounds like your one plant is too far gone to be saved, but you might wish to use a fungicide you are comfortable with as a preventative on the ones remaining so they don't suffer the same fate.

I see no evidence of PM on this begonia. I don't remember ever seeing it on mine either. Typically PM hits rexes, Mallet canes, and canes with the big polka dots. Try Milstop if you do have PM problems. It works great but the caveat is it isn't at your nursery store (yet) so buy it online. It comes in five pound buckets and is kind of high ($70 with S&H). You mix one tablespoon per gallon.

At this time in the season, I wouldn't advise you to do anything. The trees will be entering dormancy in just a few weeks and dropping their leaves anyway. Just clean them up completely and dispose of them. When dealing with woody plants like your trees, it is important to remember that powdery mildew is more of a cosmetic issue than a real threat to the health of the tree. But it is best to avoid anyway, if nothing more than to maintain an attractive appearance. PM often tends to be a late season issue in that is most common in late summer and early fall. This is when environmental conditions are prime for its development. Soils are at their driest and warmest and the air still and humid, creating an ideal breeding ground for the pathogen. And plants located in more shade and in closer proximity to neighbors can be more affected. Make sure your trees have adequate light and air circulation. You may be contributing to the problem with your watering program. Sprinklers that throw water above ground (as opposed to something like drip irrigation or soaker hoses) increase humidity. And watering in this manner 3 times a week for 20 minutes may be unnecessarily often but with the effect of insufficient water reaching down properly into the soil profile. It would be much better to water less frequently - perhaps only once a week - but for a longer period of time. And in a manner that delivers water directly to the soil at the root zone. Low and slow :-)) I can't tell you exactly how often to water - that depends on your soil conditions and weather - but even once a week may be more frequently than necessary for trees established in the landscape. Ideally you want the soil to be just moist to the touch throughout the root zone, Not too wet and not too dry. You may have to do some experimentation and testing to get the timing correct. And proper mulching can help to maintain soil moisture between waterings.