For example, is a person missing a leg, still responsible for 5 times salat a day? Or a person with mental illness - is he still responsible for any religious duties that a normal person has, or do different rules apply for people with disabilities?
different people have different duties placed on them as a muslim although the majority have the same a person with a missing leg is still very capable of performing salat or prayer
Narrated 'Imran bin Husain (may Allah be pleased with him): I had piles, so I asked the Prophet (may Allah's blessings and peace be upon him) about the prayer. He said, "Pray while standing and if you can't, pray while sitting and if you cannot do even that, then pray lying on your side." Bukhari
so from this hadith we are told to pray in all circumstances that are possible
Al-Hasan ibn Muhammad said, “I entered upon Abi Zayd Al-Ansari, who called out the Adhan and Iqamah while he was sitting.” He added, “a man advanced and led us in prayer. That man was lame whose leg was injured in the Cause of Allah, the Exalted” (Al-Baihaqi).
this is a specific example of a man who was injured and still performed prayer
but if prayer is beyond somones mental capability and it is absoloutly impossible for them to perform it then it will not be required fro them to pray allah has created these people and knows what they are capable of and will not force them to perform what he knows they cannot