Women are the violent one in only 3 to 4% of domestic violence cases (97% to 96% domestic violence perpetrators are male).

Depending on where you look, you can get surprisingly different numbers. In terms of the number of attempts, it would seem women are as frequent as men, but their attacks are underreported because they are usually physically weaker and cause fewer consequences. In terms of how often they attempt it, the numbers may actually be dramatically different.

It is far more likely that the mother stayed at home than the father, so in cases where the children are still very young and stay at home with the mother, then of course the young kids should stay with the mother after the divorce.

This should indeed be taken as an argument in any legal battle over custody rights, and the father, if he wants custody, should demonstrate that he has the necessary skills. But I am reminded of the commenter here who mentioned that in Nebraska (?), 97% of the legal decisions favored the mother; I find it hard to believe that, in 97% of the cases, the woman really was the best parent. There could very well be a "women are always better for the kids" prejudice skewing people's perception (a prejudice that has of course nothing to do with feminism, but is a prejudice nonetheless).

Men are less likely to want full custody, and in a lot of cases shared custody (say if the mother wants to move hundreds or thousands of miles away after her husband leaves her for a younger woman) isn't an option, so the mother will get full custody rather than just joint.

But I assume (please correct me if I'm wrong) that the stats cited are for the cases of legal battle, i.e. in which the men did want it?

Anyway, there is a good point here: those who defend men's rights to custody should also provide statistics about how often men actually want such rights. If this is not so in 50% of the cases, then a low or even very low number of cases benefitting the father might be at least influenced by there being relatively few fathers who want that.

Men are far more likely to have a criminal record when coming before a judge in a custody battle.

That is probably true (though I'd like to see that stats on that one as well.

Men are far more likely to have gotten caught cheating before they come before a judge in a custody battle.

That is probably also true, but it is much less compelling that the previous claim -- since cheating doesn't necessarily make you a bad father.

I don't think the idea is that men are equally vulnerable to attacks in the sense that a woman physically attacking a man would just as often imply danger of death as it does the other way round. I think the idea is that men are (or may perhaps be -- again, where are the stats, and where do they come from?) attacked as often at home. How lethal the attack is is not the issue -- I mean, if a woman were physically abused by a family member but in a way that was not clearly lethal, would you argue that she shouldn't be seen as a victim of domestic violence?

Not the lethality of the attack, but the attack itself and its objective, define the issue. Being slapped by your wife is also domestic abuse--just as being slapped by your husband is, even if he is careful not to use his full strength.

And let's not forget the possibility of using instruments to decrease the difference in strength -- knives, guns, sticks. Or also the enlistment of other family members to do the job. Also psychological manipulation, verbal abuse, ...

For an example of a paper on physical assault initiated by women, here's a URL I found a few days ago. I'd be interested in the opinion of others about it.

http://www.menweb.org/battered/fiebertg.htm