I have a level 11 druid with a lion animal companion. He gets Pounce, Grab, and Multiattack. If I pounce and get five attacks (bite, 2 claws, 2 rakes), do I apply my STR mod to the rake damage?
Yes, your lion's rake attacks gain its STR mod as a bonus to damage.
From the AONPRD on Rake (emphasis mine):
A creature with this special attack gains extra natural attacks under certain conditions, typically when it grapples its foe. In addition to the options available to all grapplers, a monster with the rake ability gains two free claw attacks that it can use only against a grappled foe. The bonus and damage caused by these attacks are included in the creature’s description.
First, note that the rake attacks are claw attacks. Claw attacks are primary natural attacks that gain your full STR bonus to damage.
From the AONPRD on natural attacks:
Most creatures have natural attacks (attacks made without a weapon). These attacks can be primary or secondary attacks. Primary attacks are made using the creature’s full base attack bonus and add the creature’s full Strength bonus to damage rolls. Secondary attacks are made using the creature’s base attack bonus – 5 and add only 1/2 the creature’s Strength bonus to damage rolls.
The second emphasized part of the description of the Rake special ability may cause some concern, as it implies that the bonus should be listed in the animal's stat block, and no such bonus is listed for the big cat animal companion. However, this concern is negated by the fact that no bonus is listed for any animal companion's natural weapons; because animal companions increase their strength score as their master gains levels, their bonus changes over time. Thus, animal companion stat blocks only include the base damage die for natural attacks, and leave calculating other bonuses to the player.
This can be seen in the animal companion lion's stat block listing no bonus for rake (or bite, or claw) attacks, while the monster lion's stat block does list bonuses for all of its natural attacks, including the rake attacks.