the father is not listed on the child's birth certificate. there is only information about the mother. after the wedding, the wife wants to take the Canadian husband's surname. do I need to change the child's last name too? or would it be better after the marriage to give both the name and the child the last names? the marriage is concluded in Kazakhstan.
-You may need to check whether the father (if known) has any 'residual' rights in KZ even if not listed on the birth certificate. (I think you should still apply for the sponsorship without anything but you should be aware that IRCC conceivably could come back and ask about this). This would be a question for a local lawyer familiar with family rights, really. (In some countries the father even if not listed on birth certificate can - hypothetically - claim or sue for parental rights. In other countries if a parent is not on birth certificate, they have no rights vis a vis the child).
-You do not need to change the child's name beforehand but it can be done. If you wish to do this BEFORE coming to Canada (in KZ), it would be better to complete the name change before applying. You'd simply use whatever name change process is used there.
-If you don't want to wait for that, it can be done in Canada relatively easily after arriving and becoming PRs.
-It won't make any difference if the child has a different family name than the mother/stepparent (except for having docs to show parental rights, eg birth certificate and name change docs).
-As a side note, if inconvenient to change the mother's family name before immigrating, it can be done fairly easily in Canada after arriving. Up to you.