I would be pro-active and take steps to file the order yourself. The care of your daughter, both physically and fiscally, is the most important thing here. Nobody in this situation sounds particularly mature or able to make objective decisions in the best interest of the child...loathe as I am to say it, better to have a third party oversee things.
You can, in Texas anyway, require that he complete parenting class(es) before he gets any overnight visitation as part of the order--you may be required to do so as well. Since he is paying support, he could righfully claim that the child DOES have a car seat--the one you use--and that he contributed to buying it.
You can also include clauses about the consumption of alcohol when he has custody of the child.
Unless the orders state otherwise, he can appoint an agent to act in his stead (i.e. girlfriend picking him up). He does have to inform you...and likely in writing....if he is to have someone else act for him so you are sure to hand over the child to the correct person. You could probably request that something is written into the orders, but remember that it will affect you too. What is good for the gander is good for the goose.
In many states, there are exceptions that can be made against overnight visits for children under three, but it isn't written in stone.
There are plenty of families where dads don't know the first thing about changing a diaper or bathing a child...and they live in the same house! Cut the guy some slack, give him a chance to know his daughter, and enjoy the little break it could give you to take some online classes or something.
Odds are that he'll lose interest...four kids can be a lot to handle. Playing house is only fun for a while. Reality eventually sets in. You'll come out better if you play nice.
P.S. Depending on his income and how your states calculates support, $45 may be exactly what you would get with an order. However, you'd also be entitled to help with other expenses (i.e. medical).