"falls" = "in case". It can always be substituted with "wenn", apart from the connotation: "Falls ich mich erinnere" means you're not sure if you will ever remember.
"wenn" also translates as "if", but it can also mean "once (I remember, I'll call you)" - you're sure that you will remember.
"Wenn ich die Prüfung geschafft habe, werde ich dich anrufen" can either mean "Once I have passed the test ..." (= you're sure you will) or "In case I (will) have passed the test, I'll call you" (= neutral). If you say "Falls ich die Prüfung geschafft habe" ("In case I have passed the test"), you're expecting you might have failed.
That's a valid question.
A closer translation of my German sentence would be:
Do you know if it's supposed to rain tomorrow?
So, to answer your first question directly, "sollen" adds the aspect of "has it been predicted that..." to the question; rather than assuming the person you're speaking to is a clairvoyant.
Could one simply say Weißt du, ob es morgen regnet? or Weißt du, ob es morgen regnen wird?, or is it an obligatory part of the sentence?
I wouldn't have any issues with either of those sentences, and I definitely wouldn't go as far as to call "sollen" obligatory there.
This helped me a lot. Just wanted to thank you for your time helping us German learners! I do this all the time in the Spanish course and it can be super tiring.
I would hand you a couple lingots if 1. I knew how and 2. If my lingots hadn't been changed to gems (which I not-so-affectionately call "jelly beans") and they gave me the heart system.
Agreed. As the calling depends on the remembering, and neither is happening or has happened yet, then the call would be in the future. Typically, one would say, "If I remember, I will call (you)". Also, the phrase "I call in" doesn't sound like something an English speaker would say.