1) Yes, it stops playing when I put my bluetooth headphones away (YMMV with wired or other methods).
2) Yes, but sometimes pressing play wont actually play. I also see this occurring with antenna-pod (FOSS podcast player), so I think this is Android 15 being broken, or they both use the same underlying (non standard?) audio library.
3) The sibling says yes, from checking the github issues, I think its not as "smart" as other players and simply does a fixed offset if you pause for some length of time. Honestly I always figured it ran back a few seconds due to how keyframes [sic] work in m4b files.
4) Yes, I do not have any data on my phone so I basically exclusively download my books to the device and play them offline. Sometimes I do get progress syncing issues, honestly I forget in which direction, possibly it only occurs when I finish a book offline then go online and it resets back to the last "still playing" sync point.
Sometimes I do have issues downloading a book, and it will stall out mid download. I can't say whether this is the app, the phone, the wifi or the server. It's sporadic, possibly fixed in some recentish version, haven't seen it for a while.
You can (independently) adjust the skip ahead/skip back buttons between 5s, 10s and 30s. I just leave it on 10s and mash if I have to. Before ABS I used Voice (FOSS, IMO great but I wanted library management too. Not as feature rich as Smart Audiobook Player, intentionally I think.) which only had one set of skip buttons so I never "missed" this in ABS.