OP, this thread makes me a bit uneasy, as I am not sure the advice you have been given is that good, plus I feel you have been bullied a little by some posters into taking action that may not, in fact, be in the best interests of your children.
Your children's interests will be best served by their parents having a co-parenting relationship that is as amicable as possible. Providing he is a loving, non-abusive father, it is definitely better for them to retain a relationship with him. And from what you say, he is.
His current accommodation is far from ideal BUT he has been approved for a mortgage so one assumes when he moves there will be cooking facilities, beds etc. The airbeds and pot noodle stuff is temporary, and will not damage the children in the short term.
The girlfriend stuff again is perhaps not ideal but actually, lots of people do move on, get new partners and introduce these to their kids. Most people do it at some point. You may get a new partner yourself in time. It's not abuse or a reason to stop contact. (Though personally I agree it sounds like too much too soon and not ideal - but it is pretty common and certainly it is NOT abuse).
Asking the gf's mom to babysit is again not abuse, and I think the reactions to this on the thread have been hugely disproportionate. He didn't leave the children in a crack den or on a street corner. She is not "a stranger" to him. When you split and co-parent, sadly you cannot reasonably expect to have vetted every person who ever looks after the children. You yourself may at some point in the future source childcare from people your ex does not know. That is both of your right as parents.
Your daughter's comment was worrying but it does not sound like it stemmed from anything serious and awful. You are right to have investigated it further, and to keep an eye on it - but your daughter saying this once is not a good reason to stop or significantly limit contact.
It is easy for people on here to demand that you "advocate for your daughter". They don't have to live with the repurcussions of that "advocacy" - your children do. Any action you take has the potential to significantly harm what already sounds like a troubled and turbulent co-parenting relationship further. And it is your children who will suffer if this happens.
By all means seek professional advice, and if you are advised to seek to reduce contact at that time, it's correct to do so. But don't just do it because some people on the internet who knows only a fraction of the story tell you to do it. . The ultimate long term goal should be a strong and cooperative co-parenting relationship where the kids get to enjoy positive relationships with both parents. However much it hurts, however much you privately hate and resent him, in this you have to put your children first, grit your teeth and endure it.
(I am not saying by the way that his actions have all been great, far from it. Refusing to tell you where he lives is petty and stupid. But, for the sake of the children, you need to aim for an improved relationship with him, not more conflict).