Is there a standard distance that a closet rod should be from the back wall of the closet? Is there also a minimum? If so, will all standard closet rod brackets conform to this depth?
As far as I know, there is no "standard" for this. I've looked for them. The only standards have to do with the actual closet dimensions that are specified in building codes. A common convention is 12" from the back wall to the center of the rod.
People typically put the closet rods in the middle of the closet. Minimum closet depth is 24" so that puts the closet rod at 12" too.
Certain types of clothing tend to extend past the edge of the hanger. The sleeves of jackets and dress shirts for example. So you want to have room for them. Good air circulation is also important to minimize chances of mold and mildew. Especially important if the back wall of the closet is an exterior wall.
The standard is 67" up to the top of a 1X4 ledger and 12" from drywall for single hanging. the centerline of the closet pole cup is typically is 2" from the top of the ledger, it can be lower, then the ledger needs to be wider to accommodate. That puts the shelf at 67 3/4" to the top. The 11 1/4" shelf will give the room needed to get the hanger on. This is what I learned 40 years ago from my then 60 year old mentor. But there is nobody waiting with a hammer, if is varies. My mentor would have waited though. If the closet is shallower than 24" you can reduce it to the center of the closet as close as 10" as Testor101 suggests. The dimension to the back of the door give a little room for this to work if need be.
For double hanging to keep clothing from dragging severely over the lower setting and on the floor, 42" and 84", I learned this from my own closet at home. If it is a child's room these can be varied too, say down to 40" and 80".
When planning how it will go together, have your center support bracket on hand that you plan to use. Some manufacturers make the seat that the pole sets in different heights. I have seen brackets that are only have 1 1/2" to the center of the pole as opposed to the one in the illustration above that shows 2 5/8" more or less. The former will need a 1X backer at the top were the latter will not. Most center brackets are gauged to use a 1X on the wall to get the back of the bracket in the same plane as the ledger.
To determine whether you need center support or not, for me, is a rule of thumb that I picked up over time. Any span that is over 4' for wood poles, it can be longer if you know it will not be loaded fully, 5' for metal, with the same exception as wood.
Half a hanger + the thickness of the clothes would be your minimum, but the vast majority are 12"/300mm, and most but possibly not all brackets are that size.
10 inches (254 mm) to the center of the rod, minimum.
The widest hangers I know about are 18.5", so half of that is 9.25" plus 3/4" wiggle room. Though some hanger rod brackets are ~12" (304.8 mm) to the center of the rod.