It will not be possible as for now and quite some time to come-and also past, NO trains at all between these 2. For overland there are buses and depending on how far away from IST you start, it may actually be best to fly.

In general dogs are utmost rare in TR and IF one has them in public, they should be muzzled etc. But it is so rare, that the rules for it are extremely hard to find

I understand, it's just that I have a half-breed Doberman and Turkish airlines don't take me on the plane. I just read that there is a night train from Istanbul to Sofia

Yes, but it currently does not run because of Turkish COVID measures. Nobody knows when it will restart.

And if this is your only journey, then an Interrail pass is much too expensive and you'd better buy a normal train ticket.

Istanbul - Sofia will be running again from 25.4. Sofia - Istanbul starts on 26.4. From 3.6. from/to Bucharest (only in summer):

https://www.economic.bg/bg/a/view/noshten-vlak-otnovo-shte-pytuva-po-marshrut-sofija-istanbul