This needs a great deal of editing
Let's talk about the activation requirement. I think it's a bit unclear. When you say "a PC takes damage of 7.5 per level", do you mean the character's level or the level of the spell they're trying to cast? I assume you mean character level, and that you need to have taken that much damage in a single hit that deals slashing or piercing damage. The rest of my comments in this section will be based on this assumption.
First off, that's going to rapidly become impossible to meet. At 5th level you'd need to take 38 damage in a single blow, which is virtually impossible on a normal hit, and unlikely even on a crit. Even an ancient red dragon maxes out at 22 damage in a single claw attack, or 34 with a crit. By tenth level it's simply impossible to activate this item in most cases, if I've read the rule correctly. (And in case you were thinking of having piercing/slashing damage add up over time until they reach the limit, I don't recommend that -- adding more bookkeeping for just a single item really sucks.)
Second, even if there were monsters that could deal out damage like this, you're going to quickly outpace PC hit point totals for most characters. To even have 7.5 HP per level, a PC needs to have a d10 hit die with a +2 or better Constitution bonus, a d8 hit die with a +3 Con bonus, or a d6 hit die with a whopping +4 Con bonus. This item is functionally limited to the tanky classes like Fighter, Paladin, and Barbarian.
You say you can "activate a stone as a bonus action on their opponent's turn". Bonus actions happen only during your turn. ("Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.") This item as written is impossible to activate. If you want something to happen during another creature's turn, you have to use a reaction, and you only get one each round. The reaction will need text to explain how it activates when you technically should be dying, or else remove that part of it.
The spell can be cast "on any target within 50 feet in sight of the character". Instead, I recommend just having it cast the spell you choose. Spells already specify their range and targeting requirements (such as being able to see the target), so adding those to the item itself just complicates things unnecessarily.
This is probably a bad idea because players are people
But even if the rest of this is resolved, the big problem is how this will play at the table. You're imagining that the player gets one chance to cast a spell and then the game resumes. But is that really what's going to happen? Is the player going to already know exactly what they want? Or is this going to turn into "okay hold on, I need to look at the entire list of spells for twenty minutes to pick out the one I want to cast"? There's a reason spell scrolls and glyphs of warding have to be specified ahead of time. Don't introduce an item that's likely to stop your game cold.