How to select the best doe to shoot?

Good question.  Older does are less likely to produce young and therefore provide less reproductive value to the herd.  However, it is difficult to determine age of a standing deer. You already mentioned the doe without a fawn method, but that too is difficult and not fool proof as a doe may have a fawn you don't notice, or she may have lost a fawn to predators or other causes.  Older does also might have more body fat if they are not stressed due to lactation energy requirements, and may be better for the table.  Fawns and yearlings might be considered the highest quality table fare, though they don't provide as much meat in the freezer.  Some hunters believe shooting a fawn is wrong somehow.  However, biologically populations always produces many more young than needed to sustain the herd and many would likely die anyway, thus harvest of a fawn is more likely compensatory mortality and not as important to the herd as harvesting an adult.  All these are choices that you may or may not be able to make when the decision is required.  It is good to remember that IDFG provides doe harvest opportunity when there are more deer than needed to sustain the herd and to stimulate growth rates of stagnant herds.  Harvest of females, contrary to what many people believe, can actually stimulate growth of a population by reducing competition for feed, increasing survival rates, and harvesting non reproductive females.  Most populations in Idaho where harvest of females is allowed are much more productive than those that don't.  My advice to you is don't worry too much about what the perfect animal is to harvest.  Does of any age usually are excellent table fare as long as they are properly taken care of in the field and at home.  Good luck and enjoy your hunt!