What surrounds the nucleus is the probability wave. But are the electrons constantly popping in and out of existence around the nucleus in the cloud?
No, electrons do not pop in and out of existence.
They do, however, switch between localized and delocalized states:
When the atom is undisturbed, the electron will be in the energy eigenstates of the system, the so-called atomic orbitals. These states are not sharp position states, the position expectation is "smeared out" over a region around the nucleus - this is what one calls electron/probability cloud.
When we make a position measurement, i.e. force some kind of interaction that allows us to measure the position state of the electron, it takes one of the position states that are allowed members of the orbital.
But the electron here just took another state (whether you believe that happens by collapse, decoherence or something else is not relevant - the interpretation you adhere to doesn't really influence this), it didn't "pop into existence" - it just switched from an energy state to a position state. Its existence in the energy state is, quantum mechanically, no different from its existence in the position state. Quantum objects are not little blobs in space.
The answer to this question depends on your interpretation of quantum mechanics.
According to the Copenhagen interpretation, the interpretation that is taught standardly in undergraduate quantum mechanics courses, the answer is no. The wavefunction simply describes the probability of finding an electron in a given location. When we are not directly observing the electron, we may only describe its location probabilistically.
Electrons dont pop into and out of existence, in the way we normally think of reality. However, until someone or something actually attempts to look at it, or pinpoint its location, the precise location of the electron is uncertain. In other words, we know the electron exists, even if we dont know precisely where it is at and any given point in time. Awesome, hey ?