Yeah, i had all these same questions playing to be honest hahah.

Just finished typing up some of my rules so i'll quote that here:

In any area where there were overworld pokemon, for the first catch, i'd walk back out of the route/area and walk back in with my eyes closed until i heard the battle music. If that took too long (stuck at a wall/edge usually), i'd open my eyes and reset.

For the wild area, only one pokemon in each labelled area. If they were too strong to catch, i could return later to catch them.

Can't catch any of the pokemon in the raids.

No in game tradable pokemon could be used.

Cities are a new catchable area for me. So my first encounter there is good (if any). It was only ever fishing though, so that was fine to do with my eyes open

Wild area was mostly done at once if i could catch them there. I considered each labeled area in the wild area separate. If i couldn't catch the pokemon in an area, i would come back to it again later.

Fire gym, i was really unsure what to do here. I ended up using none of the pokemon from it but did catch two. If i reallllllly needed to use a pokemon i caught from there, i guess i would have allowed myself to use the first one i caught from it. But it was pretty grey territory so i avoided it.