I think it's fair to say that if you were with someone and had children with them, you owe it both to that person and your children to make concessions for them - to go out of your way to provide for them, even when the relationship with the mother ceased romantically. Regardless of the romantic situation, you will always have a relationship (ie, an interaction) with the mother. That said, when two people decide to go separate paths but are bound by obligation it should be in the minds of each of them to not interfere with the other's chosen path.
I have my opinions on what has been presented. Really, I don't see it as such a big deal that the father has decided to involve his new primary female figure in the lives of his children. I can understand some tension in this realm, but I don't think that it is inappropriate, much less that it is emotionally insensitive - particularly some of the examples shown. His new girlfriend is perhaps a big part of his life, and having her around, or even in a car ride, is just one little part of that.
I understand the opposing opinions, although I disagree with them and my initial reaction is, short version "suck it up, there are much, much worse things out there", other people's subjective reality is just as valid as mine.
Push comes to shove, this world is random chaos. There are no rules, there is no defining order. It is up to you two, Orobas, and maybe you three or four with your respective new mates, to figure this out and make it work for you. I hope you arrive at a set of solutions that is agreeable for you, your ex, your partners, and provides a stable environment for your children. Good luck.
