Yes, that's what I consider 'self-tending'. Perhaps in ignorance but that's how I understand the term to be used.
Not sure I understand it... if the hitch is loaded there's nothing to tend. Tend out the slack. Do you mean break, as in rappel? I consider breaking and tending to refer to two distinct operations - tending to ascend, breaking to descend. I'm happy to be corrected if I'm wrong.
Generally holding on to my stick or the tree. I generally avoid holding my rope itself as I don't want to slip and inadvertently break my hitch. Sometimes I'll pull myself up by my hitch (or the carabiner on the hitch), then grab the trunk or whatever once I'm situated where I want to be. I have had my tether fall but I use a bungee to keep it tight to the tree so that's rarely an issue. Generally, I try to tend out slack as I climb; 'standing up' while pulling myself up with one hand and tending out slack with the other.
I prefer self-tending hitches as I want to be able to operate them one handed. I haven't been impressed with the few (admittedly DIY) Prussic tenders I've tried. On the ground they were great but at height I needed two hands to tend, which meant I needed to set a LB to hold myself in position while fiddling with my Prussic. Very frustrating and felt unsafe. With one-handed operation I can tend slack out faster. Pushing the Prussic up hasn't worked for me because I generally don't have anything weighting down the tag end of my rope; when I've tried this I needed to step on my rope to hold it in place while I pushed the Prussic up, otherwise I'd just lift the rope up by my Prussic instead of advancing it. I dislike stepping on ropes as I want to avoid grinding grit into the fibers, and I want to avoid getting my feet tangled up.
Glad you've got a technique that works for you. I couldn't make it work for me and went a different route.