Thrown ball kicked by runner
 

I couldn't find how to handle this in the case book for HS, so I thought I'd ask.

Play at the plate: F3 thows to F3 who drops ball. R1 slides into home and happens to kick/move the ball up the 1B line during his slide. Since I didn't see any intention in the act, we just kept playing. Coach wanted runner returned to 3rd.
What's the call here and just for my learning does it make a difference if the ball was hit after the R slide across the plate?

Coaches want lots of things that aren't supported by rule. You had the rule correct, and it doesn't matter whether the ball was touched before or after the runner had touched the plate.

Who threw the ball? Was his 'intention' to throw it to F2's mitt or let the runner kick it during his slide? We aren't clairvoyant out there, so when a coach asks if the runner is penalized for kicking his team's poor throw, you can tell him, "No, coach." and walk away.

The call here is, in no case, runner returned to third! 99% chance this is simply nothing. For this to be INT, it's got to be intentional.

Also ... how does one thow? :) And does thowing always cause the ball to return to the thower, like it did here going from the Firstbaseman back to the firstbaseman (Speedy Gonzalez) suddenly covering home... :)

Think of it this way: if R1 was stealing second, and F2's throw to was low/wide such that it hit R1 and bounced into the outfield, would you return R1 to first?

No you would not.

This is exactly the same situation you describe, just at a different base.