A father who does not have legal custody of his children cannot be forced to take care of those children while the mother, who has custody, is on vacation. And really, it would not be good for the children to be left with the father if that father actually didn't want to take care of them, but was forced to do so by some form of legal coercion. Children should be cared for by someone who actually wants to care for them. They will get much better care that way.
Yes
There are rules for all adults who have custody of children, joint or otherwise, fathers or otherwise. To find the custody rules for your area of jurisdiction you will need to contact your local authorities. Any child welfare organization will be able to help you with contact details.
Go for Bird Nest Custody. Learn how to prep for a custody challenge at Dads House. see links
She should pay you, but why is there no custody order in place? see Dads House below
It would depend on who the father is to you. If the one who does not want to sign over is your father he makes the decision if he has custody. If you are concerned about this you should talk to a teacher, clergy person, or trusted adult who can help you
see links and Dads House
Second couins
If the mother had primary residential, than the grandmother has a right to the children, but this not not preclude access rights. This is why this needs to be addressed in the custody decree. You may need to file a custody challenge to be granted primary residential. But, if you are being denied access, that's interference with custody, which could become a police matter. Access Dads House for further info and help. See links below.
kids miss their dads.
The male children would help their dads fish and hunt and they would learn from their dads. The female children would weave and help their mom cook and learn what their mothers do.
Their maturity level
Not all people hit their children.