angiekins

Level does not affect the damage, but strength definitely does!
 
Icons is one thing, at certain strengths, your icons get mutiplied according strength.
 
E.g. if you have a 9 icon , strength 85 (2.5 multiplier) = 9 x 2.5 = 23.5!

At the current time, level doesn't matter because we don't have stances.
As Karina said, when your pet's level is 50+, you can use Berserk, a stance that multiply the damages your pet is dealing by 1.5
Having a pet's level of 250+ enables him/her to train at the secret ninja school
 
But keep in mind that the pet's level is the basis for raising your other stats.
If you plan to lab your pet, then you'll get uneven stats with huge HP, decent Strength, average agility/movement, poor defence and level
In Mystery Island, either the "normal" training school or the secret ninja school allows you to follow courses and train stats which depends on your pet's level
You can train Strength, Defence, Agility/Movement up to twice your pet's level
You can train HP up to thrice your pet's level
 
Now everthing depends on your strategy: are you planning to have a pet with unbalanced stats (labbing your pet only) or train him/her to have even stats ?

You can train stats up to 2x a level. A level 3 pet can be trained up to 6 for every stat and a level 27 pet can be 54. If you train at the Mystery Island you can train 3x level for endurance. Stats also affect the pound fee. The level of your pet can also impact the faerie abilities it can learn, but with the BD revamp I'm not so sure those even mean anything anymore.
 
As for how much damage a level gives - I am unsure. I've searched around what feels like everywhere and was unable to find an answer.

After a certain point, the only thing that level is good for is making sure you can still raise your other stats.
 
HOWEVER, you should get up to level 50 so that you can have the Berserk option. It leaves you more vulnerable to attacks, but the damage you deal is amazing.

Probably getting the Juice Break avatar after playing on Zen mode for 10 to 11 hours after 2 "failed to send" scores previously