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(obviously more spoilers to whoever is still reading)

Confirming and adding to the above very helpful sacrifice comment! (I wish we had a wiki!). 

Source: only 1st play, and I have played up to Shine Town.

On leaving Nogas for the first time

The dialogue with the caravan traders is pretty straightforward; You ask them about your parents and/or CPU, and they offer their information for a price (is this always the case?).  You are given options for how you want to pay and possibly intimidate/fight. And then they tell you where to go and you can leave! 

Greyfold Map

There are sneakier (keyword here) ways to get the map after talking to the doctor. If you have the appropriate stats and you don't want to join either side, you can just walk into the Debris walker place, grab it, and leave. 

Now I don't know if interacting with them or Rusty's gang after or later in the game changes because of this tactic, but that's side quest business! Unless there are future updates to tie those choices into the main quest later.

Babili Death Arena and the power of friendship

If you must proceed with the death arena and don't want to turn into a skeleton in front of your parents, have some damaging weapons and good armor, stuff to heal, and your current active (hopefully very upgraded by then) companions to fight with you.
Don't bother leaving any of them behind (unless you maxed out).