Thanks for your question. While the Wikipedia page is an adequate description, I recommend you also look at the webpage of the Christian Science church itself to see how they describe their own beliefs to the public, Www.ChristianScience.com. And you can refer to Science & Health with Key to the Scriptures, by Mary Baker Eddy for the complete explanation and method of Christian Science healing and prayer.
Regarding mental illness, CS does teach that it, like physical illness can be treated and healed through prayer. Mary Baker Eddy, the founder of Christian Science, writes in her book on Christian healing:
The treatment of insanity is especially interesting. However obstinate the case, it yields more readily than do most diseases to the salutary action of truth, which counteracts error. The arguments to be used in curing insanity are the same as in other diseases: namely, the impossibility that matter, brain, can control or derange mind, can suffer or cause suffering; also the fact that truth and love will establish a healthy state, guide and govern mortal mind or the thought of the patient, and destroy all error, whether it is called dementia, hatred, or any other discord.
—Science & Health, 414:4
Keep in mind that this text was published in 1875, and do her word choice reflects the understanding of mental illness at that time. That doesn't change the point, however, that mental illness, like physical illness can be treated through prayer.
The question about what treatment is permitted sort of misses the point. CS is not about obeying rules so that God will heal you, or so that you'll be saved. It's about growing in an understanding of your identity as a spiritual idea of God. Therefore, each person is left to choose the treatment that they feel is best for them in any given situation. The general idea is that matter-based treatments are antagonistic to that goal of growing in spiritual understanding because they are grounded in the material rather than the spiritual, which is why CS advises against combining CS treatment with medical treatment. However, each person must choose for himself what to do. That said, many CStists do choose to rely exclusively on prayer because they've seen it work healing all kinds of physical, psychological, relationship or even financial problems.