During my last game, a player tried to have his character make a miraculous escape, only to wind up splattered instead. I'll provide some details, but the gist of my question is whether my ruling on action economy was incorrect.

The PC, a Level 9 sorcerer with the Quicken Spell metamagic feature, started his turn chained up in the bottom of a rowboat. His party was in the water trying to rescue him from his captors, but he was unaware of both this and the knowledge that the great white shark that had just taken a bite out of the boat was technically on his side.
His plan was as follows:

1. Cast dimension door, transporting himself instantly 500 feet in the air.
2. Cast feather fall and float downward.
3. Wait out the rest of the fight safely overhead.

Now, this particular player believes in an antagonistic relationship between player and DM (he's the kind of fellow who believes that when you win, you've beaten the DM, and when he's DM it's his job to kill the players as fairly as possible). As such, he refused to tell me the next part of his plan until he had found out whether the current part had worked, so I wouldn't have time to ruin it. Which meant that his plan played out as follows:

1. His character successfully casts dimension door, reappearing 500 feet over the boat. I ask if he's sure, and he insists.
2. His attempt to cast feather fall fails, and he plummets 500 feet. I prepare to roll damage, thinking that he'll hit the water. The player grumbles.
3. One of the other players reminds me that the boat is being held in place by the shark. Both the antagonistic player and I immediately realize his character is going to hit the boat, not the water.
4. Shouting ensues. I eventually lay out the reasoning below, and then roll damage. His character deals 76 points of damage to himself, the boat, and the NPCs that had been holding him hostage. This is enough to kill his character instantly due to massive damage.

Because this reasoning killed a PC, I'd like to be sure that I have a valid interpretation of the rules, namely, whether you can cast a spell and react by casting a spell on the same turn. I provided as support that you can't the following bit from the Player's Handbook, which specifically references spells as bonus actions:

You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

I reasoned that if you can't cast another spell after you cast a spell as a bonus action, that means even as a reaction. I also specified that it was specific to his turn, as it mentions, and that if he had fallen on someone else's turn it would have been quite possible to use feather fall. To drive that point home I likened it to trying to cast magic missile to hit yourself and then reacting to yourself by casting shield, which I would also disallow.
Is this right? Can you cast a spell with an action or bonus action and then cast a spell as a reaction, all on your turn? Or did this particular PC legitimately outsmart himself?

For The Record:

• In my games, players take turns in sequence over the course of 6 seconds. I understand the logic of simultaneous turns, but unless everyone agreed in advance what their actions were and then the round was resolved all at once, it wouldn't work at my table… and as you can see, some players don't even want to discuss all of their turn at once.

• In my games, falling occurs on the player's turn, unless they are forced to fall by someone else's action, kind of like movement. Not worried about this but, as the player agrees that this works (and has taken advantage of the rule before).

• By a non-coincidence, falling in my games occurs at the rate of 500 feet per round. Since the falling occurred on his turn, he went up and came down all during the same turn.