"It's raining now!"
Yes, it is needed.
This sentence comes with an exclamation mark. So the speaker was probably not expecting the rain, or it was not raining a while ago. To indicate a change of status (not raining becoming raining), 了 is definitely needed.
It is raining now. I want to take an umbrella. (I am only describing the status now. I am not saying about a change.)
It is raining now. I would better take an umbrella. (There is a change of status. It has become different from it was previously, or from my expectation.)
I am missing something here; I thought that 了 implies that something finished. The rain did not yet finish. Maybe the English translation should be that "it started raining now" given that the start has finished; it sounds awkward in English though... I am confused; Please help...
了 obviously has a lot of functions in Chinese. You are correct, sometimes it does imply completion.
However in this sentence, it shows a change from "not raining" to " raining" and unexpectedly. That's another functionof 了 you need to keep in mind
As for your completion case, let's pretend it's the afternoon and a speaker says
This would indicate completion because the speaker is talking about an earlier time, hence in English, "It rained this morning"
I hope my explanation is not confusing