I agree “developing mites” sounds puzzling! There is a simple explanation though. All rabbits have some mites in their fur, they get them from their mum when theyre kits. The buns immune defense keeps the number of mites down, and the mites dont cause the bun any trouble.
When a rabbit is under some sort of stress – not necessarily “bad” stress, it can be from getting a new home or even sth as simple as molting – the mites can sometimes seize the opportunity and multiply into much larger numbers, which is when they might start to cause the bun discomfort.
I dont know why your vet didn’t prescribe Revolution, she might routinely give Revolution in the clinic.
The Meloxicam I would guess is for itch relief. Itch is sometimes a response to pain stimuli. If your bun has irritated skin, there’s some sort of inflammation going on. The bun scratching itself could make it worse.
Meloxicam can have a short-lasting sedative effect when you start treatment. I just saw this in my bun who got meloxicam for 5 days after her spay. It made her very sleepy, but only for sth like 20 minutes right after I gave her the med. It was very noticeable though, she couldn’t keep her eyes open.