Is it more or less impossible to have mad psychic spellcasters (that is, those that are unsound of mind)?
The psychic magic rules make it pretty clear that psychic spellcasters must be in control of their thoughts and emotions in order to cast most of their spells (or even use certain class abilities) which, by definition, isn't the case for the insane.
I'm aware that there are metamagic feats and spells without emotion components that can help bypass this, but not without severely limiting the character's abilities. Therefore, is it RAW-impossible for me, as a GM, to throw a "crazy psychic" at my players without really hampering his abilities?
I think there's a differentiation to be made between mechanical and flavor madness. While the character might be unstable or unsound of mind, that isn't necessarily a disqualifier from emotion components. After all, psychedelia is a discipline and I would hardly think that someone who lacks sobriety is in perfect control of their emotions.
I think an insane psychic should be possible. To him, he's in control of his thoughts. He knows what he wants. It's just his perception of the world that's wrong.
You can give him some GM-variant version of Mad Magic that lets you cast while insane, and cite this as why his magic works, if need be. Your campaign, your rules, after all, and I think the flavor and experience of an insane psychic far outweighs any minor GM allowances you give your NPCs.
Don't worry about it :)
If you need a roleplaying model, you can use Master C'Baoth from the Thrawn trilogy by Timothy Zahn. He is in control of his (mostly psychic) powers, but is also insane (due to his heritage).
It's flavor.
Even if it was RAW-impossible ... who cares? Do what you wanna do. :)