If you are planning on building a dedicated long range elk rifle then I would suggest the 338 RUM. Elk are very tough and personally I believe that a .338 cal bullet is the minimum when shooting elk at long range.
I actually prefer the .338 Lapua but you really should use a large diameter action for the Lapua than what you can get by with using the RUM.
Recoil is very subjective but comparing the 7mm Rem Mag to the 7mm STW there is a tremendous difference in recoil for the gain in velocity/energy. I don't mind shooting a 7mm Mag all day but the recoil of the 7mm STW is objectionable to me. In fact I do not mind the recoil of the 300 Win mag or 300 Wby mag as much as the 7mm STW, I'm sure it's related to recoil velocity.
If you go with the 7mm STW I would recommend a brake (I put one on mine) for practice shooting. But I shot my 7mm STW twice without a hearing protection, the first and last time.
I do hunt with a 30-378 without a brake and do not find the recoil objectionable while hunting(just so you know that I'm not a total recoil weenie)
but a few rounds hunting is a lot different than firing 50-100 rounds in a practice session.