Fire-and-forget drones already exist. They are called missiles.

Missiles generally go from point A and then blow up point b. Are there any missiles that leave base, fly around for a bit trying to identify targets, and if they can't identify any targets then return home?

I said that somewhat tongue-in-cheek, but the line between "drone" and "missile" seems to have gotten pretty blurry with the Ukraine war featuring FPV drones holding a hand grenade that are effectively a human-piloted missile.

Some modern missiles can fly around to identify targets, but they can't return home. They can blow up harmlessly if nothing is a target, though. They can also dodge and weave the way FPV pilots do.

Obviously, "drone" has a much more expansive mission profile than "missile."

Missle is a projectile, propelled by rocket motor. Drone is an aircraft vehicle, capable of transporting explosives or whatever. There is a fat line between drone and a missile, there are different in everything: control, speed, trajectory, weight, flying principles. I don't know how you can say they are even close. Just because both fly and do boom? Then you can say there is a thin line between cow launched from catapult and a missile.

Cruise missiles are missiles without rockets.

A cow launched from a catapult is indeed a missile, if we want to be pedantic.

The term you're looking for is loitering munitions. From Wikipedia: "Some loitering munitions may return and be recovered by the operator if they are unused in an attack and have enough fuel"

You are describing cruise missiles