I'm guessing I'm being a necromancer now, but anyway...In DMC1 novel, Dante sees demons crying and even thinks for himself" Are they crying?", but he kills them anyway.So he does not care too much about if they cry or not.
I think the whole "crying" thing is being taken too literally, here. I think it comes down to, "Are they showing true remorse, or is that just fear that they're reaping what they've sown?"
He has a password nobody knows: imagine someone has an emergency and needs help, but doesn't know the password , so that person is condemned to die. Quite silly...
I imagine if someone has a problem of that particular nature, they seek out someone who can help them (such as any agent throwing jobs Dante's way, like Enzo or Morrison) and they direct said client to Dante. That being said, I think the password thing is riddled with inconsistencies. People also just walked right into his shop and won his services. He may play it off like he doesn't care from time to time, but it's obvious that he does. Example, in the anime episode 8, "Wishes Come True". When the woman with pigtails showed up, he claimed he wasn't interested...but he took on the job, anyway, and once completed, you could tell he was happy he'd done the right thing.
I understand your concept of judge, but he judges them under what law? His laws? But not even he respects what he preaches, how can convince the others to follow it? DMC media shows that too.
Not just "his" laws. More than likely the same principles Sparda himself worked under. Examples?
He hunts demons for revenge and money, he doesn't care too much about humans, he even kills them if he feels too, so...Just check anime and DMC4: he's more for the money than for protection, he just doesn't want to be seen in a bad light by humans.
Or, the reverse could be true. Just because he comes off as cold and indifferent, doesn't mean he is. In the novel he may have killed humans, but lamented having to do it, only doing it when necessary. Besides, the novel is not necessarily considered "canon". Beyond that, Dante hasn't killed humans in the games.
All of what has been said is pure conjecture. You assume he doesn't care, but the evidence suggests quite the opposite. When it comes to his indifferent attitude, think about the horrors he's been witness to, a part of. Wouldn't that make anyone jaded for awhile, until they find themselves again? Also, when it comes to his attitude with Lady in DMC4, it could just be that he's messing with her. Chances are, he was interested in the job before Trish took off with the "Sparda" sword. Just saying, it's all a matter of perspective.