Quick answers - mental illnesses were until recently taboo subjects with very little understanding of them, like in most countries I suppose. Awareness and understanding of depression, anxiety etc., are much greater than before with a lot more discussion about them. Japan is still behind western countries and the provision of mental health services is very patchy. I have anxiety problems and have been touched with the understanding that Japanese friends and colleagues have shown when things have been bad, though anecdotally Japanese doctors seem to have little understanding of medication for antidepressants.
Disabilities - generally positive attitudes towards full integration into society as far as they are able to in principle but poor enforcement of equal opportunites (that goes for ALL equal opportunities in Japan) in the workplace. Lack of support in schools for children with specific difficulties (both mental and physical) such as teaching assistants. I've heard that people with specific disabilities are pushed towards particular careers, such as massage for blind people.
Gay/transexual rights - positive views from most ordinary people, remember that Japan has had little exposure to the anti-gay bigotry of western religions. I was recently chatting with a group of mainly Japanese people, a non-Japanese guy brought up the subject of gay marriage, they all shrugged, said they were in favour of it as if it wasn't even an issue worth discussing. Even an elderly Japanese guy in the group who found the idea of homosexuality deeply unpleasant was in favour of allowing gay marriage.
Some of the politicians in Japan's ruling party have recently made noises against gay rights, but they don't seem to be in favour of rights for anyone apart from themselves!
Hope this is enough to get the ball rolling!