My players have agreed to instruct the Leilon children in fighting, to help them defend themselves. What skill would that be? Performance? Althetics? Combination- Strength (Performance) or Charisma (Althletics)?

Personally, I wouldn't necessitate a roll for this as long as they have proficiency in what they're helping to train, on the player's side. For those learning, I'd say it takes x # of hours instead of requiring rolls, but rolls might help reduce that, and would depend upon what they're teaching. If they're teaching fighting, I'd suggest Athletics or straight Strength or Dexterity depending upon the weapon.

"Fighting" is something that all adventurers are proficient in, with respect to their field.

I'd let a player use their Primary Ability + Proficiency Modifier, without worrying about a specific skill. However, if a player has a reason to use a skill they are proficient in, rather than a straight ability check, then I'd probably let them.

Beyond that, you could certainly ask for Persuasion/Intimidation checks for additional morale/motivation beyond actual skills training.

Near as I can figure, Instruction would be based on Intelligence.  You get everyone together and you explain what they need to do.  It might be better to use Charisma and Persuade them to fight first, and then your Instruction might be listened to more easily.

• Performance based on Charisma might work, by showing them the moves and letting them practice.
• Athletics kind of depends on how much str they have, so some would be harder to teach than others.
• Using Strength to push them around when they are practicing might not be well advised.
• Or did you mean maybe flexing your muscles at them?  That's what Charisma based Athletics is for.

Bards would be great for this.  Most of the more effective ways seem to be based on Charisma, and there is nothing like telling heroic tales and singing to make people believe fighting to defend themselves would be a good idea.

Myself, I wouldn't dream of encouraging children to fight.  I'd teach them how to run away and hide.  Bards and Rogues would be great for that.  Rogues are sneakier, Bards are more charismatic.

I would agree with a Performance check, on anything the player is proficient in.  I have been trained by a LOT of different people, on a LOT of different things (First Aid, Fall protection, a dozen or more alarm/access/cctv systems and more)  I can tell you that presuming all the trainers were of equal knowledge of their topic (proficient) those who Performed better, as in presented the instruction in an interesting, relevant and understandable manner, were the most effective trainers.

I would suggest a Performance roll for the player, and maybe an Intelligence roll or Athletics, for the trainees, to see if they learned well.  The higher the Performance roll, the lower the DC for the students.

I personally wouldn't use intellect. I've known a lot of really smart people who are good at what they do, but can't teach it to save their lives. If you want to force a roll, go with performance or even wisdom insight (good teachers can tell what their students need to learn a skill). If it were me though, I'd just say this action will take X amount of time to train and move on. You could even roll a d6 and say this is how many days/weeks it will be to teach them.

I'd say Wisdom. Instruction is mostly about observing your students and adjusting the lesson as appropriate to how they are going.

Charisma is your interacting-with-people statistic, so I'd use that.