Is it normal, my issues with mra's?

Both men and women's rights groups twist facts like you wouldn't believe, fuck both.

Yeah, those are all valid criticisms, although I'd point out that it's not just men's rights advocates who do those, or analogous things. They're all pretty common problems in political movements really.

This only criticism I would argue with a little bit is the contest thing. I don't think there's anything wrong with trying to leverage existing awareness of abuse against women to point out that men suffer from similar abuse. But it needs to be done tactfully, honestly and without implying the problems women face aren't also serious.

There are biases against men in the case of custody over children, making an issue about it isn't as much of a "Don't even try, bro" as much as it is a "You can try, I wish you luck, but the likely case is that you'll fail, but we're trying to change that, bro" thing.

"Domestic violence and rape".
How so? I severely doubt you take part on MRA sites and more so just snoop around there. To suggest rape towards a male is less or abuse towards men is less is part of the problem because it is just as bad, if not worse, especially when it comes to domestic violence because there is very, very little help in comparison for them, so they are more likely to have nowhere to go for help and more likely to be trapped in that abusive situation.
Also, recent studies have shown that women are more likely to be emotionally and physically abusive. The fact that you are saying it's on a lesser level than abuse towards women, for no reason other than "vagina" is why there is barely anything done to help men in these issues, so yes, stating the things the way they do is the way to bring awareness to the issue, what you suggest is what will keep it continuing the way it is. You have a very gynocentric view of things, how two can go through the same but women have it worse for some weird reason even when they are the minority.

"Misogyny".
I think this is partially due to feminists. MRAs hate feminists, and rightly so. The problem is that feminism has forced its representation over women even when women do not want them to, however they are the only ones doing it so it has stuck. So what I would assume has happened is that because feminism forced its representation over women, these people see women and feminism as mutually inclusive regardless of the fact it isn't, and so their hatred of feminism is extended on to women, which then feminists get to say "look! Musoginy!" Even when it is possible it was them that were the cause of it with their forced representation over women.

"Complaining/solutions".
You are aware of how every event they have held has been treated, right? Feminists breaking the law to prevent them and so on? The most MRAs can do at this stage is increase their numbers, which I believe they are doing pretty well, in a diverse way I may add. Groups that make change don't just happen, it takes time, so give it time.

"Hurt women".
How? Give examples. I see them saying that men should be given the same equal treatment as women while women be held to the same standards as men, so the eye for an eye part is partially true but in a way that lines up with equal treatment.