This is my first post. When a team has players showing up late, and let's say 3 players show up after the game has started and their names and numbers are not in the book. There is only a mximum of one technical for that infraction, not 3 right?

Right. One technical foul only charged to the team, no matter the number of infractions. One foul counted toward the bonus. No indirect T charged to the head coach.

Welcome to the forum.

Of course you could always tell the coach to put all his players in the book so he doesn't get penalized for the latecomers.

Summer League games, I tell coaches to make sure they are in the book before the hit the floor, doesn't matter when they get here. I figure the game is for them and I don't want to penalize them for putting down their PS2 controller and getting off the couch. Besides, coaches are laid back and wearing sandals and few parents come to games, which means games with little or no grief. (except on Mother's Day)

The funny thing is that the COACH showed up late along with the three players. The game was actually started with a parent sitting on the bench with the kids.

Can someone please verify the ruling for this other type of “book error”: Let’s say (heaven forbid) that three players on the same team are wearing different numbers than what is recorded in the book. The first time each of them checked into the game, the book would be corrected and the team assessed a technical foul, resulting in a total of three technical fouls, right? Would all three technical fouls be assessed as indirect technical fouls against the coach (which would result in ejection)? If the three players with the incorrect numbers all switched jerseys so that the shirts on their backs match the books prior to entering the game, there would be no infraction, right?

Obviously, there are prevention tools that could be deployed to avoid this catastrophic situation, but I just want to verify the ruling.

Wrong. See below
Wrong. See d. below

Rule 3-2-2 - After the time limit specified in Article 1, a team is charged with a maximum of one technical foul regardless of how many infractions of the following are committed:

...
b. Adding a name to the team member list
c. Requiring the scorer to change a team member's or player's number in the scorebook
d. Requiring a player to change to the number in the scorebook
...

AZ, once the R has approved the book, no matter how many infractions occur, only one technical foul can be assessed to a team.