I'm confused. That last picture looks like a driveway already in place. Is that yours, or your neighbors?
Most people don't realize that allot of us here in Texas have very little to almost no topsoil. If you have topsoil, then you should remove it. More then likely, you just have the grass growing on clay and rock with nothing else.
Just scrape off the grass and have some gravel delivered. It might be called limestone, cleachie or road base. The main thing is that it has rock in it that's several inches large down to fines that are the size of sand.
You need a minimum of four inches of rock for it to lock together and become solid. A little thicker is better, but not a huge difference for residential use. Less will mean potholes and movement that will lead to issues and failure.
The county, or whoever owns the road in front of your place will decide what sized culvert you need. Bigger is better, but also more money. You didn't say how big the culverts you have are, just how long. The important number is how wide the hole is in the culvert. 12 inches is the minimum, with 15 and 18 being able to handle allot more water. I had to put in a 24 inch culvert at my place.
Depending on who you get the rock from, you might also see if they have clean fill dirt. Put the culvert in place and just dump the fill dirt over it. Build this up until you get the slope you want for both the road and the sides so you can maintain it.
Before putting rock on the fill dirt, be sure to drive over it allot to compact it.
Eddie