1/ No - I would be very skeptical about the reliability of pH meters that read 0.01 and cost less than $1K
2/ No - self answering from 1/. Trust readings on a good meter to 0.1 - yes
3/ Maybe - depends a bit on what you call rapidly. As the wort pH is mostly a result of water chemistry and malt acid, most of the acid comes (90%) from bacterial lactic acid on the husk it is very available so 10 minutes or so and I think it would be fairly stable.
4/ Probably - given time. I suspect homogenise might be the wrong term, stabilise would be closer. the major effects on pH aren't from the sugars that we extract from the malt but from the water, acid, other minerals in the malt (phosphates), melanoidins in dark/roast malt, all of these go into solution fairly quickly.
We are looking to work in a pH 0.4-0.5 points wide range in what is a fairly complex environment at varying temperatures. If you can get reasonably consistent readings to 0.1pH don't rely on ATC too much you should get good results and control over your brewing. Without spending the earth, and getting an ulcer stressing over things you cant really change anyway.
If you want to get a good reference pH you probably would need to titrate, can be surprisingly accurate.
Mark