1.They're all female, which would cripple their population, unless you deviated from canon to make male Gems or some other means of letting them breed.
As dude said, it won't effect entity pops - and you can make them immortal, which fits with the show.
2.I have no idea how Fusion would work, and also unfusing.
Fusion between two gems could be difficult - all i can suggest is maybe a shapeshifting spell that you can ONLY use on others. You cannot use it on yourself. Its not quite the same, but its something.
3.They can summon weapons from nowhere. You could do this with reactions, but you don't want non-Gem adventurers doing it as well. You'd also want to limit the type of weapon summoned to the appropriate Gem type. I know the Dragonball mod does something like this, and it heavily relies on DFhack. I don't know much about how DFhack works.
Create a reaction, and make whichever Gem Weapon they summon require a specific material, which that caste of gem will have a natural ability to conjure up - meaning that only a Ruby could make a Ruby Dagger, for example.
4.They can shapeshift into a variety of forms, or change individual parts of their bodies. This also doesn't heal them, which makes it harder to replicate properly. You also don't want the player spammed with "Urist McGem has transformed!" messages.
Well, this is another case of make do with what ya got. But really, shapeshifting is much easier - quit complainin!
5.Like you said, the poofing/regenerating/bubbling system. Really, just how they take damage in general. Gems seem to take an enormous amount of punishment before ultimately being brought down in one shot. They also lack vital organs. Dwarf Fortress kills typically revolve around ripping people apart until they die of either blood loss, respiratory shutdown, extreme pain followed by an easy killshot, or attacking the head. Gems would ignore most of these.
Make them constructs, like Bronze Collosus, but with a single organ, the Gem. Make it so they can be knocked unconscious(this will represent poofing), and by attacking the area where the Gem organ is located(for now, we make it so all gems are in the chest), you can shatter them, killing the creature.
6.Their bodies are weird. You'd need to do a lot with the "Holograms with mass" tissues so they don't die easily, but also aren't overpowered. If you wanted to replicate the random placement of gemstones on their body, you'd have to make a ton of nearly identical castes and gem bodyparts. That's doable, but difficult and time consuming.
See above note.
7.The monsters would be easier, but still run into many of the same issues.
Once again, above note applies.
8.Lapis and Peridot's powers involve manipulating available materials in the environment, which I don't think is really possible. Peridot's limb enhancers could probably work as an item syndrome from DFhack I think.
Well, Peridots metalkinesis is kind of unique - that is, without being deprived of enhancers, and without actively TRYING to obtain said powers, she would never have had them. So we can kinda ignore that. Lapis - well, she has flight, thats easy enough, make her breate under water, and give her a Material Emmision power for water that shoots it at a high velocity.
9.If their clothes are part of their bodies, you'd need to deal with the Gems being upset about being "naked" unless you gave them no emotions, made them all immodest, or gave them something to wear on their lower bodies.
Meh. Give em a loincloth, no biggie.
10.Dwarf Fortress combat relies heavily on dismemberment, and I don't think you could cut limbs off a Gem without poofing them. Maybe if you carefully reworked all of their tissues and body parts to be immune to severing?
Once again, very durable, but can be knocked out with enough strikes. Limb loss should rarely come up it they're as durable as they need to be.
11.The real Peridot wouldn't be in it and that's terrible
She kinda annoys me. So im not bothered. Anyhow, you can always start the game as a Peridot, and roleplay THE Peridot, so it works.
All problems can be solved if you arent obsessed with 100% accuracy, and settle for 90%.
But seriously, Regular Show mod. Am i the only one here who thinks it should happen?