Oh my...the whole point is: it is just plain rude to bring someone to a party who wasn't invited. Period. End of story. I am so surprised that people don't know the basic rules of etiquette and, basically, how to be polite. That is societal norms.
It is not okay to drag along whomever you please just because you think they should come. If you can't or won't get a sitter, then decline the invitation. As a hostess I would feel obligated to include the other children that you brought, (because that is what a good hostess does, makes everyone feel welcome) even if you "kept them separate" and at Chuck-E-Cheese that is an extra $19 a kid! Now please imagine that each kid brought 1 or 2 siblings!

I'd have to mortgage the house!
As for the invitations at school. All boys or all girls I would think was a thing more for, like sleepovers, or Libby-Lou, or American Girl parties, things that are gender specific. As for handing out the invitations in school, again, that goes back to basic manners. While we realize that the invitations are not being handed out in the middle of class, they are still being handed out in front of other children. In the cafeteria, in the bus line, on the playground, etc. It is rude to do so. Extending an invitation to ony certain members of a group IS rude. Just mail the invitations. ...and we all know how children are...they are carrying around the invitation, they are talking about it...jeesh...they are kids!
On another level, your children have a right to have their own lives and interests, and that includes choosing their own friends, and spending time with their own friends. This is how they develop independence. Constantly ramming siblings down their throats is not fair. It is also not fair to the younger siblings to make them think they are intitled to everything the other siblings are. Children have a right to have something that is just their own...including friends. Again, if you can't or won't get a sitter, decline the invitation!
The time to teach children about manner is when they are young. By teaching them to drag along uninvite guests you are teaching them rudeness. By teaching them to openly shun certain people you are teaching them rudeness. This may be how some people act, but it is not what is socially acceptable.