Pman brought up good points. High power greens are everywhere and it's too easy to not know what you're doing with something not-so-safe.
As for diode power, 532nm lasers actually use an infrared laser diode at 808nm which is mostly invisible light. It's converted through crystals to visible 532. Many 532 lasers (certainly the 303) won't have an IR filter so use caution. Even when it seems "dead" and no green light is produced, you can still be shooting out IR. Some people can see the IR as a very faint deep red. DON'T look into the aperture. You can look at it from the side and sometimes see it. Or point a camera at the laser and look at the cam's screen.
The diode itself can be running several hundreds of mW, and only a fraction 50-100mW gets converted to 532. Along with maybe 10-30mW of IR (these are ballpark estimates). Just always be careful with 532 lasers even when they seem dead. Also as the battery drains and the green output decreases, you can still be getting a lot of IR.
You would need an LPM to find out the real power output of the light itself. Keep in mind this reading would include the IR. You could put an IR filter between the laser & sensor and get the power of only the 532.
All the common Lithium-ion
rechargeable batteries are the same voltage. Listed as 3.7 but will be about 4.2 at full charge, decreasing to about 2.75V when "dead". This applies to all the common sizes (18650, 18350, 16340, 10440, etc). The difference is capacity. An 18650 holds a lot more than a 16340 for instance. But it also depends on the current draw. I have a low power 650nm which uses a 16340 and I almost never need to charge it. I have a 300mW 532 which draws 1500mA which burns through a 16340 in 20 minutes.
Try to avoid cheap unprotected Li-ion cells. Also avoid cheap chargers. You can get a Nitecore i2 on Amazon for about $14. Worth every penny.
Your Laser 303 is a decent laser but don't be surprised if it fails. It's just the way cheap greens can be. They are mass produced to an enormous level.
Enjoy! :beer: