I don’t know enough about Czech grades to comment, and I generally agree with you about the common unfairness of custody disputes, but the rest of your claims seem a bit shaky.
Your point #2 (along with other reasons for a child to more commonly live with a single mother, e.g. the father isn’t known, is in prison, or runs off) partly explains your points #4 and #5. From the document you cite, 27% of abuse victims were living with a single mother and another 5% were living with their mother and another adult, while only 3% were living with a single father. If you only look at children living with both parents, it looks like either parent could be the abuser with approximately equal likelihood. Looking at the source document, your claim #5 seems to be a complete misinterpretation of the data, but in any event, the largest cause of fatalities is “neglect”, and most fatalities were in children under the age of 3.
More generally, I suspect a significant proportion of children spend substantially more time with their mothers than their fathers, e.g. because fathers are more likely to work, tend to work longer hours when both parents are working, and do much less housework and childcare. I suspect these disparities are even more prevalent if you restrict your examination to households where child abuse occurs.
If you want to understand child abuse by parents, you should try to gather concrete data about the family structure and other context. Aggregate statistics about which parent is more often the aggressor can be misleading without considering those other factors (most relevantly here, who the primary caregiver is).
Interpreting this document to imply that children are at much greater danger from women than men is disingenuous, or sloppy at best.
Similarly for your point #3. What is the context for the “attacks” you are talking about? Gang violence? Bar fights? School bullying? Muggings in dark alleys late at night? You don’t cite a source, so it’s not clear quite what you’re referring to.