Mine is at Coolatai, started out just as you describe, was plenty good enough for most STOL planes, I did my training of it and it was described by Phil Martin from Superair as being in the top 3% he flies super off. So I tried to make it better
Be very careful grading that you don't form gutter or banks that concentrate the flow of water and make the peddles wobble if you get out of line when you are landing. I graded mine myself and left a bank of dirt well off to the side of my cross strip which acted to gather and divert a significant amount of water straight down the middle, a second bank on the other side wouldn't let it off until I busted out about 200 m further down. A fair bit of consideration of letting water follow its natural flow is required. You don't need a crowned up road to land on, it may be more weatherproof but will take a lot more skill to stay up on top of, then give you a lively ride when you don't succeed.
As far as grass variety goes, I haven't found anything that doesn't form tussock, is drought proof, frost proof and grows all year round, so just mow the native grass, which has proved it will grow there, and just mow it low to keep it smooth.