The striker hits the ball forwards, very slightly to off, but mostly up. The bowler makes his/her way towards the area where the ball will land in order to catch it (before it lands). The non-striker is headed in the same direction and running along with the bowler (on the bowler's right), on his/her normal path to the other end. They touch, but the bowler catches the ball anyway. Should the non-striker receive a warning or does he/she have some right for his/her path towards the safety of the other end or is this require something else? (The non-striker claimed that he couldn't be at fault, because he didn't see where the ball was.) And what if the bowler missed the catch, judged likely due to the faint touch or distraction?
Neither party has inherent right of way. The bowler is not obliged to change his/her path to avoid the batsman and the same is true for the batsman. In the situations you describe, the relevant Laws would be Law 37.3 Obstructing a ball from being caught, which comes under the obstructing the field mode of dismissal.
The striker is out should wilful obstruction or distraction by either batsman prevent a catch being made.
In the scenarios you outline, the contact, and any obstruction, is unintentional, and therefore there would be no penalty or dismissal (other than the catch, if it was made).