Once a beer is carbonated in the bottle, are there any benefits to aging it?

Beer was "mild" when young and "stale" when old. Stale bread is "bad" but stale beer used to be regarded as good. It still is but we don't call it stale ("stood ale"?) any more - we call it bottle conditioned and it fetches a premium. One local brewery released a batch of bottles over 2 years old that it found at the back of the warehouse. It was very short lived and immensely popular!

Beer changes considerably over time in a bottle. The change is usually for the better. One month is OK for the impatient, but try a beer at 6 months to see how bottle conditioning can change its quality. Personally I would never (willingly) drink a beer I made that is under 4 months old.

You've got 5 or 6 gallons of this stuff! Go ahead and try one!

I almost always try one after 2 or 3 days. It's often quite enjoyable in the first week. Once I drank about 12 bottles in the first week -- and enjoyed every one of them!

You can pop one open now, and it's a good learning experience to keep drinking your beer regularly so that you can see how it develops. I know, tough life! And you'll probably find like I do that the beer is at it's peak when there's one or two bottles left.

8 days may not be enough time for all the CO2 in the headspace to dissolve back into the beer, so you might hear a loud hiss but not have much carbonation. After another week, the beer will be better carbonated. But don't let that stop you from having a taste now. Waiting for beer to be ready is one of the hardest parts of being a brewer!

As to how long to age the general rule is higher OG = more ageing required. Apart from wheats, which tend to age quickly (usually best at 3-5 weeks depending upon gravity) most 1.036-1.048 beers will be at their best around 4-6 weeks or more. Higher OG beers require more time still, e.g. a typical 1.060 beer is best left for a couple of months before thinking about drinking it regularly. But still, open a bottle every month, to see how it's developing.

Agreed with above poster. I made an American Amber ale with 4 whole vanilla beans added in secondary and bottled it. I tried one 2 weeks after bottling and their was a bitter taste I didnt expect and the vanila after taste was so pronounced as to make a negative impression. I thought I had wasted 5 gallons of beer. I waited another week and tried again, and the tastes had mellowed. After a fouth week, it is one of the smoothest, most drinkable beers I've ever tasted.

I always open one at 7 days, just to test carbonation.

As far as aging, yes, it can change a beer significantly. I've had beers really come into their own after 2 weeks, and after 4 months. There is no hard-and-fast rule here.