Yes, Muslims can divorce. Here are different terms of divorce for men and women but divorce is permitted although it´s not recommendable in Islam. After divorcing, man is in responsibility to take care about his children.
------------------------------
Maintenance of children is a duty of the father, according to scholarly consensus, whether he remains married to his wife or divorces her, and whether the wife is poor or rich. She is not obliged to spend on the children when their father is still alive.
...
Maintenance of children includes providing accommodation, food, drink, clothing and education, and everything that they need, on a reasonable basis, depending on the husband’s circumstances, because Allaah says (interpretation of the meaning):
“Let the rich man spend according to his means; and the man whose resources are restricted, let him spend according to what Allaah has given him. Allaah puts no burden on any person beyond what He has given him. Allaah will grant after hardship, ease”
[al-Talaaq 65:7]
...
It is permissible for a divorced woman to ask her husband for payment for breastfeeding the child, according to scholarly consensus.
http://islamqa.info/en/89708
----------------------------------
So long as the woman does not remarry, the custody of a child belongs to the mother until the child reaches 7 years of age. A boy is given the choice to choose between his parents when he turns 7.
There are some differences of opinion on who gets custody of the child if the mother is remarried and the girl when she reaches 7.
http://islamqa.info/en/91862
http://islamqa.info/en/171456
I wish you´ll get also other answers from members whose know more about his matter than I.