The 7-08 is a good round. Don't get me wrong. But the ammo isn't available everywhere. A lot of places, but not everywhere. If you were a realoader, i'd say gopherit. But, there really isn't any noticeable difference to me in the .308 and 7-08 on recoil and there sure as hell isn't any difference on game. Both are capable for all listed.
The .30-06 will be bit more but certainly not a heavy recoiling round. The stock on the T3 doesn't fit me for squat so, that wouldn't be my choice. I'm assuming you've fondled a few to decide which one fits you the best? In a Tikka there is ZERO advantage to get a .308 over a .30-06 since all the actions lengths are the same. Plus, in the 06 you can shoot 130-220gr bullets and not skip a beat.