If the one who does not pray denies that prayer is obligatory, then he is a kaafir according to scholarly consensus. If he affirms that prayer is obligatory, but he does not pray out of laziness and carelessness, then he is also a kaafir according to the correct scholarly opinion, because of evidence that is well known, some of which has been quoted in the answer to question number 2182.
Based on this opinion, the zakaah, fasting and Hajj of one who does not pray is not valid, and it is not permissible for a Muslim woman to allow him to be intimate with her until he repents and starts to pray.
The wife has to pay zakaat al-fitr of behalf of herself, and if she pays it on behalf of her children that is fine.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Zakaat al-fitr is obligatory and like all other obligatory duties, each person is obliged to do it for himself. So you, O man, are obliged to pay zakaah on your own behalf, even if you have a father or brother. Similarly, the wife is obliged to pay zakaah on her own behalf, even if she has a husband. End quote.
Fataawa al-Shaykh Ibn ‘Uthaymeen, 18/261
Based on this, you should pay zakaat al-fitr on your own behalf. As for the children, if they are young then you should pay it on their behalf, and it does not matter if that is done without the father's knowledge.
But if they are adults and have reached the age of majority, they should pay it on their own behalf if they have wealth.
And Allah knows best.