The centrifugal force is a ficticious force, meaning it's not caused by real interactions but by the presence of a non-inertial frame of reference.
When you are on a merry-go round and feel the centrifugal tug, there is nothing pushing you. It's your inertia that would like to make you follow a straight line, instead of rotating along with the non-inertial frame of reference.
(a) What force counteracts this attractive force to prevent all of the
electrons from sticking to the nucleus?
The force only needs to be counteracted if the system were in equilibrium. Which is not the case. Forget quantum and think of the planetary analogy: the Earth orbiting the Sun. There is a net force on the Earth, which is sustaining the centripetal motion.
(b) Why is the kinetic energy of the electron twice the fall in
potential energy? As an example, a meteor falling towards earth would
not stop when the KE is twice the PE, it would continue falling till
all the PE is converted to KE simply because the attractive force does
not stop at any point.
Because the electron (the Earth) is bound to the nucleus (the Sun).
Bound systems subject to power laws obey the Virial theorem, one of the results of which is the relationship between potential and kinetic energy that you mention.
(c) Why doesn't the electrostatic attraction ensure that the highest
probability of finding the electron is on or within the nucleus?
It does, for the lowest energy state.
The radial part of the Schrödinger equation (for the Hydrogen atom, for simplicity) gives you the first solution that goes as $\propto e^{-r}$. So it does peak at the centre. For higher states, the ones with non-zero angular momentum $\ell \neq 0$, you have energy coming from the rotation as well, which somehow balances the attractive Coulomb potential and results in a maximum at $r \neq 0$.
However, what the Schrödinger equation gives you is a wavefunction, which (when modulus-squared) is a probability density. You have to multiply it by the volume of a particular region of space in order to get an actual probability.
And close to zero, the volume of a zero-centred sphere is vanishingly small, killing the chance of finding the electron there$^\dagger$.
(d) Given the Pauli exclusion principle saying that all electrons
cannot have the same energy, meaning all cannot be sticking to the
nucleus or within the nucleus, still there has to be an opposing force
holding the electrons at successively higher energy levels, like the
reaction force from the tree holds the apple up at a higher potential
energy. If the opposing force is not there, then they must fall and
lose energy. Why is this opposing force not needed for an electron?
The Pauli exclusion principle says two identical fermions cannot be in the same quantum state. In this case, what you mean is that they cannot have the same energy, spin, and angular momentum quantum numbers.
Hence, they cannot all be in the same state. They stack up to higher energy levels.
No repelling force required. When you fill books in a bookcase, and fill up a shelf, you move up to the next shelf. There is no repelling force making you do so.
But fair enough, is the Pauli exclusion principle a force?
(e) Can we say that while an electron is in an atom, it does not
behave like a particle at all, so it is not attracted by and does not
accelerate towards the nucleus because only particles can do that and
waves cannot, meaning there is no such thing as a charge, attraction,
mass or acceleration for a wave.
The electron is behaving like a matter-wave. If it were a particle, it would be an accelerating charge and hence would be radiating energy away, and its orbit would be decaying. Making atoms unstable.
Forget the particle/wave duality. That's just used for intros to QM.
Electrons, atoms etc. are described by a wavefunction which satisfies the Schrödinger equation, and has an associated dispersion relationship. I call it a matter-wave. You can call it whatever you want.
$^\dagger$: think of it this way. The highest probability density of me finding a girlfriend in the middle of the Pacific Ocean. Because that area has the lowest record of negative mentions of me on the internet.
Great.
Now multiply by the number of women in that area.
Less great.
