I think that you’re getting a lot of information here. Some of it is general, some of it specific, so you’ll need to pick and choose what is applicable to you. I think that you’ve seen, however, that folks here who do this sort of design work everyday (raises hand) are more apt to ask you for a plan or a photo before getting too specific. That said, we let’s start in general concepts.
I know that terminology can be regional (and I’ll be using New England terms here), but in general “gravel” refers to a mixed gradation of mineral soils from smaller than sand up to cobbles. Typically the larger stuff is screened out. And you don’t want too many “fines” (silty particles that pass the 200 sieve). Ideally, a well graded gravel will give you the best structural base, because the finer gradations move in to fill the voids in the larger gradations. Think of a gravel road as your typical country “dirt” road.
What a lot of people call gravel (small, angular and relatively uniform sized stones) is called “crushed stone”. The railroad ballast mentioned earlier is a good example of that. You have to be a bit careful when working with all one sized stone, although the angularity usually ensures that the stones will lock together. You can get crushed stone in mixed gradations, just like gravel. Something we use around here is “dense graded crushed stone”. If you want a well constructed gravel road, it is hard to beat dense graded when placed on a gravel base. In lieu of dense graded, some people are turning to reclaimed pavement; the result of when pavements are milled down prior to being overlaid with new pavement.
This may help to address what your road should be constructed with, but there are other design factors to consider here that will keep you from rebuilding your driveway every time it rains.
Your road should pitch in a minimum of two directions. It should pitch from side to side, or if you prefer a crown, it should pitch from the centerline to either side. And your road should have a pitch along the length of the road. In general, 2% (1/4"/ft) is desired.
Stand in the middle of the driveway. Is the ground level about the same on each side? If so, consider a crown. If one side is decidedly higher than the other, consider just pitching it from high to low.
Now look down the length of the road. Is there a low point in the road? If so (silly as this sounds) make sure that’s where the culvert is. Otherwise, water will just pond at the low point.
Now you need ditches. I’d try to go 18 inches deep at a minumum. They serve a few purposes. If the ground on either side is higher than the driveway, you’ll need a ditch to intercept storm runoff, or it will run across your driveway. And even if the ground is lower on one side of the road, you’ll want to pick up that water if it looks like it will flow off your property and flood out your neighbor. Ditches served to channel that water parallel to your road and bring it to the low spot. And, if you’re in areas subject to frost, it helps to lower the water table under paved surfaces to avoid frost heaves.
So now you’ve got water flowing down either side of the road to the low spot, and into a culvert. 12" isn’t a bad size and will handle quite a bit of drainage; I like 18" as a minimum (it passes more debris that will clog a smaller pipe), but it’s real easy for me to spend your money. Obviously your ditches are going to have to deepen here, because your culvert should have some cover (depends on the pipe you use, could be from 6-24 inches) and you want your ditch to meet the bottom of the culvert. This also will increase the headwater depth which will help get the water through the pipe, provided it has somewhere to go once it gets through.
From your description, it sounds like the water just equalizes on either side of the culvert. All dressed up with no place to go. Unless you provide a positive relief for that water, all your hard work will be for nothing. You need to get that water flowing from the culvert away from your driveway and to the stream you mention. Again, without knowing how much water we’re talking about, 2% is probably a good rule of thumb.
The trouble with the generalities here is that I don’t know the pitch of your driveway, the grades on either side of your driveway, how much water shed is tributary to your driveway that will have to be handled by the culvert, and the amount of elevation change to get down to the stream. It could be that everything is damn flat there, and that always causes drainage problems.
I am not your civil engineer etc. etc.
Good luck.