1. No, except as outlined in #2.
2. No, except distances shorter than 1 mile.
3. Hell No.

Doesn't count unless it's posted somewhere. If I'm pretty sure that the mat is "as-published" I might count a split of a longer race.

Swimming is a different story as the pool is the only thing that is a truly known distance.

1. Sure, if I think the time/distance are accurate it's a PR
2. Sure but tracks are faster
3. No way, too many variables on a treadmill

In summary if I cover the distance and feel it's measured correctly... PR.

1.10k road run PR set in a training run=No, not a PR. Too hard to know if the distance was accurately measured. Measure a training run with 3 different GPS devices and you will get 3 different measurements. Civilian GPS is no as accurate as everyone likes to think. So (IMHO) a 10k road run PR has to be set on a USATF certified course, in competition.
2. 10k track run PR=Yes, but it is a track PR, distinct from a road PR. Track PRS are going to be faster.
3. Treadmill run 10K PR=No way, not a PR.

Other kinds of races, like Olympic Distance Tri PR, get a little more difficult. Too many measurement, terrain and variables involved. But it you are honest with yourself and others, you won't claim a PR at a race that you know some of the segments are short (or were shortened). I mean, I know guys who have done a race advertised as an "Olympic distance tri" that had a 500 yard swim, 23 mile bike, and 5.7 mile run, and they claim that race as their Oly Tri PR. In their heart, they have to know they are lying to themself and to others but, heh, whatever floats their boat I guess....

Runners only count prs in races.