How likely am I to a) end up with fluid all over the bike and floor, or
b) kill myself on the next steep hill?
It’s impossible to bleed them without spilling a bit of fluid, at the very least. It only takes one mistake and you have fluid everywhere. Do it somewhere where it will be easy to clean up. Shimano fluid won’t hurt your bike unless you get it on the brake rotors or pads.
The concept of bleeding is simple; you’re just filling the system with fluid and getting all the air out. In practice, it can be fiddly. When you manage to get all the air out, the lever will have a solid feel. As long as you close the bleeder on the caliper and on the brake lever, fluid can’t escape and air can’t get in so they aren’t going to suddenly fail as you’re riding along.
However, making sure the system is properly filled, with no air in the reservoir, is important. If you have air in the reservoir, the brakes can go spongy if you turn the bike upside down. Generally, pumping the brakes then holding the lever for a few seconds, then releasing the lever will help the air escape back into the reservoir and get them working again, but it’s much better to make sure they are properly filled in the first place.