I can confirm that the power-up related achievements can be earned without buying the power-up packs. There is a way to earn free power-ups in this game. In the Rise and Swine chapter you need to collect desserts for the King Pig during the levels, and feed them to him. He will sometimes ''burb'' power-ups for you.
1) Start the Rise and Swine chapter in the game.
2) Complete the first level of this chapter. (Make sure you pick up some dessert during the level)
3) You will see an icon with an crown appear, when you have completed the level. Click that.
4) It should bring you to an screen where you can feed the desserts you have to the King Pig.
5) Then just drag the desserts to the King Pigs mouth and sometimes he will give you power ups
Few pointers. As i found out, you can complete one level multiple times to get the desserts in it. Good level to grind desserts is the fifth level of Rise and Swine. Use the puncher to punch the Pig trought the map.
It also appears, that when you have completed one level from the Rise and Swine chapter, there will be cake appearing in the other chapters too.
Big thx to Bonien2, for helping me with this :)