Is fishing allowed next to the housekeeping camp site?

It'd be hard to actually catch anything here because of all the activity, noise pollution, rough terrain and current.

Bait fishing is not allowed, flies and artificial lures only, and all rainbow trout in Yosemite Valley are catch and release only. I would avoid fishing near camp regardless, there are scheduled raft trips down the Merced that start near Half-Dome Village and pass through the stretch that is next to Housekeeping, so you'd have better luck elsewhere anyhow due to how much human activity there is on that stretch of water.