Are some species pokemon more prone to certain illnesses or conditions than others, or even certain types? Could Psychic type pokemon perhaps be more prone to mental problems? What examples can you give?

Absolutely, anonymous. In actuality, though, psychic-types are often inherently stronger and more mentally stable than most other beings in existence, purely because of the burden telepathy and empathy hoist onto their users. Thus, it’s like any other form of Darwinistic evolution: those who couldn’t adapt to the psychic strain also couldn’t pass their traits on to offspring, whereas those with well-fortified minds could.

However, ghost-types tend to be susceptible to mental illnesses, and it’s not for the reason one would think. You see, some ghost-types, particularly yamask and phantump, are supposedly the deceased spirits of human beings. Regardless of whether or not that’s true, it’s undeniable that they are more melancholic than most other pokémon, and some are even prone to symptoms that uncannily resemble those of either PTSD or depression. They aren’t the only ones, either. Members of the shuppet line are also prone to these issues, and spiritomb is supposedly aggressive due to the exact nature of its unfinished business, so to speak.

Fairy- and normal-types are also prone to mental issues in the form of anxiety, likely due to an inborn aversion to humanity. Clefairy in particular seem to be affected by this, but anxiety has also appeared in audino, meowth, granbull, and skitty.

As for physical afflictions, there are indeed a wide variety of possible illnesses that affect some pokémon and not others—far too many to go into, to be frank. For example, grass-types are far more susceptible to cold and may experience frostbite sooner than other pokémon, electric-types that have been exposed to electromagnets or extreme surges of electricity may experience shock sickness (a condition in which they’ve stored too much electricity and/or can’t release any of it), and steel-types can actually be burned by their own armor under extreme heat (a condition called self-burning). In terms of diseases, some infections affect some grass- or bug-types but not others, some mites infest the fur or hair of some mammalian pokémon but not others, and the flu may devastate some bird-like pokémon but not others. I wouldn’t say there’s any one type or species of pokémon that are more susceptible to disease or conditions than another (even though I’ve mentioned grass-types twice in this paragraph alone), but there are certainly plenty of examples of certain conditions afflicting some pokémon more than any other.