It is indeed gravity which keeps the atmosphere around the earth and which holds us to the ground. Gravity is the force which attracts bodies toward each other due to their mass. The strength of this attraction depends on the masses of the bodies and the distance between them through:
F = G*(m₁ * m₂)/r²
(G is the universal gravitational constant, m₁ and m₂ are the masses of the bodies, and r is the distance between the centers of mass of the bodies).

Astronauts on the moon "bounce" because gravity on the Moon is so much lower than that of Earth (by a factor of ~6) that the astronauts were strong enough to jump and fling themselves around simply through "normal" (for the surface of Earth) motions. The lack of atmosphere has nothing to do with this bounciness. In this video, an astronaut on the (International Space Station) ISS is in an environment with an atmosphere but with low (essentially zero) effective gravity. Similarly to moving on the Moon, moving requires a sort of bouncing rather than standard walking motions that work on Earth.

[Interestingly, part of the reason astronauts fell over so much when on the Moon is because gravity is important for the sense of up and down. The low gravitational field of the Moon meant the subconscious parts of the astronauts' nervous associated with balance couldn't decide which way to try to point their heads.]

No - the reason why astronauts on the moon bounce around is because the moon's gravity is about one sixth as strong as Earth's gravity.

The gravity of an object depends on the mass of the object. Earth is bigger than the moon, has more mass, and thus more gravity.

Gravity is what keeps the atmosphere from boiling off into space, but the amount of atmosphere is a little bit more complicated than just gravity. For example, Venus has a much thicker atmosphere than Earth, but about the same amount of gravity. Still, without gravity, the atmosphere would boil off.

In short, you are correct.

The gravity of a planet or moon depends on the size of the planet or moon. Earth is big, so it has stronger gravity than the puny moon. Earth's gravity pulls us and the gas molecules toward Earth, forming the atmosphere and keeping us on the ground. You're right.

Here are some answers from Quora that have 'upvotes':
from Quora "The atmospheric pressure and gravity are very closely related. The force of gravityis acting on mass of gas particles in the atmosphere and pressing towards Earth's centre. Gravity is a binding, pull force. ... Our weight is also due to gravity and always pressing downwards."