I had a bed very similar to this, the slats didn't fall through but they were too unsupportive/cheap.
Also a lot of your creaking is most likely from the frame as much as the slats. Here's what I did.
Firstly I purchased these for $14 AUD,http://maxilock.com.au/
They're Australian so I don't know what you could get over there, but you want something that attaches your siderails to your head and foot board with some tension, not just suspension like most cheap brackets.
Secondly, I replaced all the slates in my bed with 13 pieces of 89mm x 19mm boards from my local hardware store and cut them to size.
Thirdly, I screwed them all down. I used self tapping screws with lagging. The lagging section is important and should be greater than the depth of your slats. This allows the slats to be pulled down hard against the bed frame instead of just suspending between the screw and creaking. You'll need at least 39 so buy a pack of 50.
I used a cordless drill, and started from the top of the bed, on the left side of the slats, screwed it down, moved to the middle, screwed that down, then to the right. Then I repeated that 12 more times, and I labelled the slats 1-13 and marked the right side with permanent marker so for disassembly all the holes would match up. Also make sure your bed frame is dead square before you start and recheck after a few slats.
Best of luck.