The Swede is a fantastic long range cartridge and has many sure-thing recipes to make it work quite well.
7mm bullets are indeed used by competitive long range shooters and the UK F-Class team won the FCWC with them, but it is definitely not the most popular caliber, even though it probably is pound-for-pound one of the best for competitive shooting. It requires a big case to work well and these hurt the shoulder and the wallet. Bingo 1010 has done some great things with a 280AI (Necked-up, blown out 30-06) and I must confess he has me tempted to go there myself.
6.5 cartridges still offer superb long range performance, wind-bucking and accuracy. Best of all, the 6.5 Swede will not kill you with recoil, and brass from Lapua is the best you can get.
The Swede is a fantastic cartridge that requires no fiddle-farting: Add powder, primers and bullets. I would definitely build if you plan on trying to shoot more than 500M. You won't find many factory rifles that can shoot that sort of distance well. The gun HAS to be accurate if you want to learn anything from your results, otherwise you'll never know if the 3 foot group was you or the crappy factory barrel.
Firemachine: all the cartridges you mention (Assuming you mean a 260 in 6.5) are based on 308 cases, so brass is identical. Bullets are more expensive the larger they get, but powder use is negligible.
Recoil increases with bullet weight and size. 308 being the worst. 260 and 7-08 are progressively less. The smaller the bullet in a given case, the less barrel life though. The Swede has roughly the same recoil as a 7-08. It is not punishing to shoot at all.