Good questions. She will either need permission from the father for them to immigrate or court ordered full custody.
To be clear, usually three family members petitioned at the same time, do go through the process together. The issue about coming together or not is governed by the fact the visas are only usable during the six months following the medical exam. If you mean a couple months after mom arrives, no problem. Both or you or just mom can go get them, if you don't want them to travel as unaccompanied minors.
If you plan to do it yourself, start by clicking on the word Guides at the top of any page here. Study, then become an A-Student of the I-130 and it's separate instructions.