What's an appropriate shelf-life for refrigerated pre-peeled garlic?
According to one farm (Christopher Ranch), theirs lasts 7 weeks from the packaging date, and has a "best by" date on it.

I doubt this is identical for all sources, but it seems like a decent baseline.

As for botulism, I'm pretty sure what you've heard was related to keeping peeled garlic unrefrigerated in olive oil, which used to be a common practice, and is dangerous.  Here's the original report on the incident that started it, and this article has more info on safe handling practices.

From my experience, it'll start to turn slimy and lose its texture and flavor near the end of its shelf life, and you won't want to keep using it.  But as for health risks, you don't have much to worry about as long as you're keeping it in the fridge.