The first thing that is crucial to understand is that feminism itself is not the enemy. In fact, intelligent feminists should care about the problems you list. I'm not saying they do, but they should, because feminism isn't anti-men, it's anti gender roles. This means that problems like men being pressured to bury their emotions, or men not getting proper mental health care, or men not being taken seriously when they are raped, are all things that feminism is supposed to be fighting against. The fact that our society often ignores these problems is not because of feminism, it's because of sexism. Feminists, often do ignore problems like this with men, and those feminists are either ignorant or bigoted, because sexist oppression caused by society to men, and sexist oppression caused by society to women, have the same root and are part of the same problem that feminism is trying to fix. This is not a fringe view of feminism. This is the normal view.
The second thing that is crucial to understand is that the caricature of sexist tumblr-warrior "feminists" (that do indeed exist) are a distraction. The fact that people like this are taken seriously probably bothers me as much as it bothers you. However, these people are misrepresenting feminism. They do not say what feminism is supposed to represent. You can think of them as the crazy bigoted Christians or Muslims that are technically part of the same group, but do not represent the whole.
Finally, you have to realize that gender roles and sexism are oppressing men and women. I think the reason so many feminists react badly to men's rights advocates is because they marginalize women's problems while promoting men's. In truth I think this is just a reaction from seeing feminist groups do the reverse. Really, both are a problem. Just because one is a problem doesn't make another problem any more or less important. So instead of saying "This is ridiculous to be worrying about when much worse is happening to men and nobody cares," try saying, "This is an important problem," and separately, "These other things are happening to men and nobody cares."