What trash does Burmy/Wormadam Trash Cloak use in order to make said cloak?

That'd make some sense, but that would partially make it's cloak edible. It would also attract at least one Pokemon in some cases. Multiple, depending on how much leftovers-to-dust ratio that it has & what type of food that it's using. Guessing that the cloak matches the color of said food, this would mean that discarded Pecha Berries, Persim Berries, Mago Berries, Nanab Berries, Qualot Berries, Magost Berries, Watmel Berries, Payapa Berries, Colbur Berries, Kasib Berries, & Kee Berries (kind of a stretch, since that one is more of a salmon & cream color) are all relaltively viable candidates, with all but two of the berries, Colbur & Kee berries, being more than displeasing to any Pokemon who doesn't like anything but sweet berries (I'm assuming that the most liked flavor in Pokemon nature is the sweet flavor, not trying to get too far into it with Natures). Other than that, there are Slowpoke tails, Poffins (pink ones, which, if you know, are usually the sweetest kinds), Sweet Pokepuffs, the inside of a Big Malasada (Is that sweet too?), etc, etc.

It's a big ramble, but nonetheless, I'm just trying to say that anything food like could possibly make it one of the easiest targets for Normal/Flying Pokemon to munch on. A foody cloak is possible, but rather harmful in practice.

And what's with the rods on Burmy Trash Cloak?

I'll be honest, I can't argue against that, nor could I prove it. It could possibly not exist, but there doesn't seem to be evidence that they don't exist. Therefore, proving it is impossible at the time, and disproving it is plausible, but not possible.

Bravo.