This is just to be a discussion about hit dice (and the hit points gained from them). I'll start by stating a few premises about hit points (and thus hit dice) as I see it. Of course many of you might not agree with some or all of these premises, but for the sake of discussion I would appreciate it if you respond in agreement with them and thanks.
1. Watsonian answer: Because Barbarians train their physical bodies more, and must have a greater amount of luck or preternatural survivability due to the fighting styles they employ. Their connection to nature, primal spirits, or other vital forces also tends to inure them to the slings and arrows of outrageous fortune.
Doylist answer: Because the game is balanced around some classes having more HP and others having less. If you gave Barbarians a lower hit die and Sorcerers a higher hit die, Barbarians would no longer be nearly as worthwhile to play, especially because Barbarians pretty much always have to fight in melee, while Sorcerers almost always fight at range if they have a choice.
2. See above. There may be Watsonian reasons in either direction, but the Doylist answer is that it's a balance consideration. You would be powering up Wizards (the class that least needs additional power) and weakening Barbarians and Fighters (the classes that most need attention to not fall behind). It would absolutely be unfair to punish small races by giving them smaller HD without giving them anything else in the exchange, and I'm very surprised you even ask the question (unless it was purely to see what people would say) because...yeah that seems pretty blatantly unfair to me.
3. Again, balance concerns. 4e actually allowed HP to be calculated from other stats...but only if you spent resources to MAKE it calculate from something else. I believe the backgrounds
Auspicious Birth and
Born Under a Bad Sign were the main ways to do this, which gave up the alternative benefits a BG could give in order to give you a few more baseline HP. Didn't affect healing surges though, so it was of limited effect in the grand scheme. Just letting it happen, without any effort or resources invested, would again seriously and deleteriously affect the balance of the game. Dexterity is already a god stat, granting initiative, a bunch of skill benefits, AC, one of the "strong" saves, higher to-hit, AND damage (the last two if using a finesse weapon). Letting it ALSO improve HP...why would anyone specialize in anything but Dexterity after that? You'd be creating an obvious dominant strategy, that's clearly Not Good, and Constitution would become a dump stat for most characters.
4. I could see temporary hit dice, as in the healing mechanic, being acquired via magic items. Actual permanent hit dice should not be tied to things that can be lost, stolen, or passed to other party members because that could create serious bookkeeping headaches that are not worth the effort. Permanent or semi-permanent (that is, they
can be lost, but not easily) boons or enchantments, on the other hand, would be fine. I am leery of just outright granting extra HP
especially from magic items. Perhaps THP instead? Maybe call them "innate" THP, so you have the "innate" layer tracked separately from the regular THP layer to avoid issues with either not stacking or accidentally making
all THP stack.
5. Given my answer to 4, yes, but again I am leery of such things being tied to the relatively flimsy and porous connection of "a magic item I'm wearing/wielding/using."