First of all, wow, this game it wacky. It's so satisfying to smash those bulbs!

So I beat the first boss (the pig) once, and I'm not sure how. I just smashed all the objects on got on his head as fast as possible, but when trying to do this again, I found that the items they give you don't allow you to do so in the time limit... Winning this level can't be so random, can it? What am I doing wrong.

Also, After getting to the second level, I tried to access it later from the main menu but it was still grayed out... do you have to beat a level to have access to it again?

Yes, it is kind of random beating anything, as I said in my review :( You can help yourself a bit if you've got a hard piece of trash to smash the pig into when it's first given to you. Remember that with the next pieces of trash you should only be smashing the solid pieces into it, carefully lay bouncy balls to the side and throw sponges and fish to the side because anything covering the piggybank totally protects it due to the game's janky physics.

You just have to live with the spongy stuff... and hopefully burn it later if you are on a level with fire.  Otherwise, try to break it all before piling up.  Rotate a knife to point down and you can break a dumbbell pretty easily.