How does 4-part harmony work?
 

I've recently become interested in acappella songs and harmonizing, but I lack any music theory training and thus have very little technical knowlege.
For this question, let's assume we have to write or perform a 4-part harmony passage. I'm talking something within the scope of 'normal' Western pop/rock... music (not something avant grade, etc.), and no beat-boxing, singing vocables like oohs and aahs or ad-libbing.

I understand where the first three parts should go: one is the melody, and the other two some type of harmony. That leaves us with the fourth part.

Since singing any other note that is not in the first three parts will result in clashing tones between at least two parts (if I'm not wrong), the only thing that's left is singing octave of either the meoldy or one of the harmony parts. Is that correct?

Is it common for the fourth part to shift between more than one parts (for example, you start singing an octave below the melody, and then shift halfway to singing an octave below some other part)?
Is it a common practice in vocal arranging to make one of the parts periodically discordant to achieve a certain effect, even for just a passing note or something? Is this more common when there are 4 parts?

Hopefully, my questions are clear enough. Thanks for bearing with me!

In many vocal traditions (such as the Barbershop Quartet) it's pretty common to have the melody switch between parts, so one part won't necessarily always have the melody.

The melody itself tends to be made up largely of chord tones, and if not suspensions and passing tones which don't really sound that dissonant in context as the ear hears where the melody is heading to. Certainly the melody will double one of the other parts at times but often all 4 parts will be describing a chord (such as 1, 3, 5 and 7). Beyond getting the melody and harmony in place there are many other considerations, such as avoiding parallel 5ths (unless using them for a deliberate effect) and getting contrary motion between the outside parts.

To do this really well, study baroque counterpoint. There are tons of books on this and most of the examples tend to be written out in four parts. Modern four part writing is based on this.

This answers my question; thank you!