Unlike most spells, prismatic spray (and prismatic wall) can do a number of different damage types depending on dice rolls. But how many different damage rolls would you actually make? Do you just roll 10d6 once, and apply that same damage to all damaging rays? Or do you roll 10d6 separately for each damaging ray that hits?
You should roll damage once for each beam.
Chapter 9: Combat in the Player's Handbook (and the Combat section of the Basic Rules) has the following rule under the heading "Damage Rolls" (emphasis from original source):
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them.
It uses the examples of fireball and flame strike, which are simple since they produce a single effect which hits multiple targets for the same amount and type of damage. In other cases like eldritch blast, magic missile or scorching ray which produce more than one ray, beam or projectile, each one deals the same amount and type of damage. In those cases each ray, beam or projectile only hits one target, even if more than one of them hit the same target.
Prismatic spray is an unusual case, since each beam produced can potentially hit multiple targets, and each beam does not do the same amount and type of damage - only the same amount (for those beams which have a damaging effect). However the multiple targets hit by a beam would take damage "at the same time" for the purposes of the rules (i.e. in the same instant/action the beams are cast), and the same amount and type of damage too, so it seems reasonable to roll damage once for each beam - i.e. each target hit by the same beam uses the beam's single damage roll. (For ease at the table, I'd have all the affected targets roll to see which beam hits them before rolling any damage.)
Prismatic wall is more straightforward - each layer of the wall is a separate effect, so the damage is rolled separately for each layer. This happens on each creature's turn, when they try to pass through the wall, so the damage would be rolled separately for each creature affected.
To answer your last question: both the sorcerer's Empowered Spell metamagic option and Lore bard's Cutting Words feature work as normal, though the former is a little ambiguous. Empowered Spell says:
When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum one).
This is the exception to the normal rule that you have to spend the points and use a metamagic option when you cast a spell, but it doesn't address whether it applies to a single or multiple damage rolls. I would say you could split the re-rolls across multiple beams, but it's not going to have a big impact on more than one, since at most you could re-roll five of the dice, and so it's true that it won't be as effective for prismatic spray as for a spell with a single damaging effect.
Cutting Words isn't going to make a big difference to the damage taken from prismatic spray unless the caster rolls really badly, but that's true of its effect on the damage rolls of most high-level spells. That fits the flavour of the feature - it makes sense a bard's distracting words would reduce the damage from a sword thrust better than they would a magical explosion.