I asked some experienced folk and got the following opinions:
Possible, but difficult and might need help. I’d suggest going via Liberty Springs, then following the ridge over to Flume.
It’s doable and dogable. {My dog} has done it a couple of times. More stress for me then him.
Yes it is dogable. {My dog} and I have done it. The thing about the Flume slide trail, where it difffers from Huntington for instance, is that it is possible to bushwhack any section where you don’t feel like going up over the rocks, so a person or dog should never feel stuck in some particular place. As we went up I noticed herd trails all over the place.