laser pointer class III and potential eye damage

First and foremost, you should know that the legal restrictions on laser pointers are not enforced. Therefore you should not assume the class labelling is correct. One study found that 90% of green lasers in the US had an illegally high power level.

You also should not assume the power rating is correct. It says 5mW probably because that's the most desired power level that consumers want: As bright as possible without being obviously illegal. But really it could be 2mW, or 10mW, or who knows.

Every laser pointer is dangerous, is not a toy, and should not be pointed anywhere close to other people ...but it takes extreme and unusual bad luck to cause permanent eye damage if its labeling is correct and it is really under 5mW (Note: I am talking specifically about visible-light, continuous-wave green lasers here). Here is a literature review, and you'll see that some injuries were indeed caused by laser pointers with less than 5mW of power, but most were injured by much more powerful lasers, like 30mW and higher.

Best is to find, buy, or borrow a light power meter. They're not expensive, and they're fun to have anyway.

Finally, stepping back, the review cited above found 111 patients with eye injuries from green laser pointers. Given how many green laser pointers are around, I wouldn't consider owning a legally-compliant green laser pointer to be a notably risky activity, compared to other day-to-day risks.

Owning a super-bright one (e.g. 30mW or higher) is something I would consider awfully risky. Treat a super-bright green laser pointer just like you would treat a circular saw: You store it well out of reach of children (and out of reach of irresponsible adults!), you leave it unplugged / remove the batteries when it's not in use, you don't use it if you're drunk, you don't have fun with it, etc. etc. etc.

The classification is dependent on wavelength, and the potential for damage. I would heed the classification, it is right on the edge of safe.

If you want a safe green laser buy a lower rated one.

Realistically it is unlikely to cause damage unless the beam is incident directly into the eye, but damaging the eyes with lasers is pretty much irreversible and will result in vision loss (usually just partial loss). Lasers in the visible range damage the retina, which is what makes them so dangerous.

Personally (and I work with lasers often) I would use the one that you purchased for Astronomical pointing (i.e., star gazing) only. If I wanted to do something else I would get a lower power laser, it isn't worth the risk of eye damage...