Do I need to pre-drill holes in particle board for screws?
To answer your question: it depends.  The larger the screw, the more likely you'll need to pre-drill.  If you're installing the screw near the edge, or if the screw is larger than maybe 1/8" wide on the solid part, then I'd pre-drill.  To pre-drill, make the hole almost as large as the solid part of the screw (not the threads themselves, they need to have something to bite into).  Also, make the hole as deep as the screw minus the size of the bracket.  Use some tape on the drill bit to see when to stop.

That said, for your bed, looking at the plans, I think you'll get the most added stability by adding a short piece of 2x2 vertically as a center brace under the metal bar.  Check the hardware isle of your HI store to see if you can find an adjustable foot to put on this board, and then attach it via a few screws to the center beam.  This is actually a quick way to detect some lower quality beds that use wooden slats from side to side.  If the slats have a center brace going to the floor, it's a lower quality bed.

Edit: For clarity, just imagine a table with a 5th leg right in the middle.  You have a long span from the headboard to the foot of the bed which will benefit from an extra support directly in the middle.  The adjustable legs I'm thinking of screw into the wood leg of a piece of furniture and look something like this (this is a cutaway view, you simply install it into a hole you drill into the bottom of the leg):