Quantum model of the atom

what is meant by a probability cloud of electron(s)? does this mean that the electron is rapidly moving inside the probability cloud/ orbital, present in some regions for a longer time than in others, or is it sort-of smeared throughout the orbital until it is observed?

Neither. There isn't anything underlying the probabilities in Quantum Mechanics. This is in contrast to say, usual probability where we use probability to deal with things we do not know about the system. It's not like we are saying "the electron could be here, we just don't know." Rather, we actually cannot say anything about the position of the electron until we measure it. Therefore, this throws out the idea that it stays in some regions longer than others since it isn't in any region before measurement.

The idea of it being "smeared out" also doesn't make sense. Electrons are point particles; you can't smear them out.

if an electron exists as a stationary wave around an atom, then what is "waving"? mechanical stationary waves, for example, consist of displacement oscillating with amplitudes that vary with position. What physical quantity is actually oscillating here? Surely, the electrons themselves aren't moving up and down, are they?

This is a mix up with the idea of the wave function of the electron. However, the electron itself isn't the wave function; the wave function just describes the state of the electron. The electron isn't existing as a wave itself; electrons are particles.

We do not know what the probability cloud actually is, all we know is that if you look for the electron, the equation for the cloud tells you how likely you are to find it in different places. That's how it gets its name. There has always been a lot of argument about whether it might be an actual particle or something sort of smeared out or something much more weird, so the convention among physicists is to simply ignore the question. If you are taught "probability cloud", best to settle for that until your teacher tells you otherwise!

A "stationary wave" is also often called a "standing wave". It means that the wave appears not to be moving, even though it is shuttling back and forth over exactly the same region all the time. The vibration of a guitar string is a more everyday example. But again, the thing that is doing the waving is that mysterious whatever-it-isn't that turns up somewhere-or-other.

Let's first discuss your first question because it forms the meat of the confusion.

The probability cloud of an electron is simply the function $vert psi(mathbf{r})vert^2$ which gives you the probability density at $mathbf{r}$ corresponding to the probability of finding the electron in a given region of space upon a measurement of the position of the electron.

This probability, while very similar to the probability that shows up in classical statistical mechanics, is different than the classical statistical probability in the following crucial sense: in classical statistical mechanics, when one would talk of the probability of finding the electron somewhere, they don't mean that the electron is actually not at any particular place before the measurement. In quantum mechanics, we do mean that. So, your guess that the electron spends more time in one place than the other would make perfect sense if we were talking about probabilities of a particle in classical mechanics. However, in quantum mechanics, that is not the case. In fact, the electron simply doesn't have any position unless you measure it. So, it is also not true that the electron is somehow spread out or has multiple positions at once.

The crucial insight of quantum mechanics is that one cannot associate all physical quantities in a well-defined manner with a particle at once. Certain physical quantities are incompatible with each other, this means that if a particle has a well-defined value of the first quantity, it doesn't have a well-defined value of the other quantity. What we do have is the probabilities for the different values of that other quantity that we will get when we measure that quantity for the given particle. The most famous pair of incompatible quantities is the pair of momentum and position. A quantum particle cannot have a well-defined position and a well-defined momentum at the same time. However, for a free quantum particle, the energy and momentum are compatible so a free particle can have both a well-defined momentum and well-defined energy at the same time.

So, in conclusion, the probability cloud is what it is, the probability cloud. And that probability is different from the classical probabilities in the sense described above.

Your second question is much simpler to answer. I will reproduce what I wrote in one of the comments:

"Wavefunction" is simply a name, don't read into it too much. Take the actual definition of the wavefunction more seriously than its name. If the electron is in an energy eigenstate, nothing physical about the electron would be changing at all with time (the wavefunction would pick up an overall phase factor (?−??? where ? is the energy and ? is time) but that is physically irrelevant and no measurable quantity depends on this overall phase factor.).