Does the Double Slice feat cancel the penalty for Power Attack damage on secondary weapon attack?
It's preetty fair (and comfortable for calculating =) if it does.
And also: what about secondary natural attacks (bite attack when fighting with two weapons)?
Despite the name "Power Attack" and despite the fluff that says you sacrifice accuracy for strength, the Power Attack feat does not increase your strength nor does it ever say that the extra damage is "strength" damage.
So technically it is not your "Strength bonus".
Since Double Slice lets you apply your full "Strength bonus" to attacks that normally would not, and since Power Attack does not in any way change your "Strength bonus", you would not get any synergy with these two feats.
In short, no, Double Slice does not cancel the "penalty" on Power Attack damage.
There is no way for most characters to avoid the penalty to power attack. Double slice only affects damage from the strength mod, and power attack cares very, very little about your strength mod (and that is only relatively rare natural attack side rule when you get x1.5 str mod).
Monks and brawlers are cool though. They can even get x1.5 power attack if they 2 hand, even if they only get x1 str mod, since that is part of power attack's language.
So dip brawler, I guess? They flurry actually works like TWF, and counts as TWF for feats and upgrading the flurry. So 2 levels of brawler let you avoid the problem. And you can just go with a single weapon, if you want (having 2 is kind of wasteful, actually).