I must have missed that excitement but its the usual aftermarket lure of bigger = better.
The ECU doesn't start pulling timing until 45 deg intake temps, so there's quite a bit of headroom in the OEM intercoolers. The S3 one seems to hold a flat temperature curve of 10-15 deg above atmosphere over spring/summer, and around 5-10 above over winter. On hot summer track days, an aftermarket one would probably work out better but it depends on the boost pressure I guess. I paid £260 for the S3 one from an Audi dealer, so they're not that expensive. That was about 6 years ago though!
Yeah 350 is about right. That is the number the CDL engines used to run in the championship race cars, although Borg Warner only rate the K04 to 330hp max, so there may have been some engine tweaks on the race engines, such as different cams, intakes, exhausts etc as to not require as much boost pressure.