DSG is a dual clutch gear gearbox, I don't know what they put in the Accord but Id bet decent money it isnt a dual clutch system. Take a quick butchers on Google if you want to know the techy side of it.
Something as big as how a car changes gear is a very personal thing. Some people swear by the DSG but I didnt take to it, much prefer a manual gear change, its all down to what you like (try and test drive the DSG though to get a feel and an idea)
There have been some issues with the DSG box which have ended up being pretty costly. I cant for the life of me remember if VW replaced it, I think they gave a token gesture towards the cost, maybe someone else will be able to help. The fault in questions was with the mechatronic unit. Might be worth Googling that to get a better idea of the problems. However Im pretty sure if the car was going to have had the problem, it would have had it by now so chances are its either had it replaced and sorted or will be fine (please dont ever quote me on that).
There are quite a few common problems (as with any car). Early models had issues with DV valves splitting (this was sorted in later models). Mine had issues with the coolant system which a few of the other lads have problems with, but wasnt shockingly expensive (I was lucky and had mine sorted by my VW warranty). There have also been quite a few steering rack issues (take a look at this tread regarding that -
http://www.golfgtiforum.co.uk/index.php?topic=32092.0)
Just make sure you do all your usual checks on service internals, cam belt / water pump changes etc.