Do Champion Herord gain Life and Damage for each Champion level?
Or is it just about rewards and bragging rights?
Yes, their health, match damage, and ability damage all rise a bit with each champion level.
Definitely yes.
For 3*, you won’t gain much power damage until they are around lv240-ish. For 4*, you will notice the change right away. (All due to the fact that power and health start to scale exponentially around lv240-ish.)
off to tips and guides
I do it mostly for the champion rewards, since I don’t really ever use 2* and 3* characters when not required to. But the rewards are quite nice, especially near the end! Plus I win them with much higher frequency than 4* and 5* characters.