Who is on strike after a catch, and does a run-out attempt on the non-striker count as a ball?

Both are set out in every CGL 2025/26 playing conditions document, following the 2022 Laws:

After a catch (Law 18.11): the new batter takes strike, even if the batters crossed or the dismissed batter wasn't at the striker's end, unless it was the last ball of the over. No runs count from the ball on which a catch is taken.

Non-striker backing up too far (Law 38.3): if the non-striker is out of their ground from when the ball comes into play until the bowler would normally have released it, the bowler may try to run them out. That's a run out if successful, credited as a run out rather than to the bowler. Whether or not it succeeds, the ball doesn't count as one of the over. If it fails, the umpire calls dead ball.