After a TO, ROP is used for both.

Not after a TO, if A1 isn't at the spot for a throw-in, put the ball on the floor. If A1 is not available for a FT, it's a T. (And, yes, ther's plenty of game management to do before we get to either of those results, of course)

You do if it’s after a timeout or intermission (NFHS).

First off, I did look, and don't deny that I'm incorrect. Rules clearly state that you do use ROP after a timeout or intermission. Now all that said, why? Can you (or anyone else) help clarify this for me? Why place the ball on the ground and begin the count? And the reason I ask is because the way I read and interpret the rules, the free thrower is not allowed to be outside of the semicircle until after the release (9-1-3-e), so the moment you place the ball on the floor (at the thrower's disposal), then they are simultaneously violating, and as this is an offensive violation, the ball should immediately become dead. I don't see a provision in the rules that allows the shooter to enter the semicircle and attempt the shot. What am I missing?

Timeout ...
 

NFHS: Request, and be granted, a timeout will avoid a ten second violation.

True, but I'll go back to the original quandary, this should be considered an immediate violation on the shooter the moment it is at his disposal, so I don't know how you can call a timeout if a violation would occur first.

In the ROP situation, the violations for not filling the required spots (for both teams) are changed from being a T to being a potential FT violation for both teams. There isn't so much as rule that states so but for the ROP procedure to make any sense, it has to be that way and I believe there are case plays clarifying that.

If the defense isn't in the lower spaces, you give the shooter the ball. If the shooter misses, a violation is called and the shooter gets a new shot. If it were not for the ROP procedure, that would be a technical fall as soon as the defense doesn't take the spot. Likewise, if the shooting team isn't coming out, you put the ball on the FT line and start the 10 count. At any other time it would be a T for the shooter to not come to the line.

Avoiding the need to call a T makes sense, and a delayed violation for the defenders also makes sense if they are not on the block in time. But I still can't justify the ball on the floor and count for the shooter. (And yes, I guess I can justify it by saying its in the rule book, but I don't see how you can set aside the lane violation here.) You had referenced a case play, and I'm sure it was from years ago, but if anyone can find one, I'd appreciate it. Again the only part I can't resolve is if the shooter himself is not there in time, and why that isn't an immediate violation for not being in the semi-circle.

From the 2011-12 book (the numbers may have changed). It is similar, but not quite the same. It does hint at the thrower not being in the semicircle. The only way that happens is if you put the ball at the thrower's disposal without the thrower being there. The only time that can happen is after a timeout.

Fair enough. Appreciate you finding that-