B) My cell phone was not working properly, so it needed to be repaired
SO is a bit like THEREFORE (which is a bit more formal).
it shows that what was said follows logically from what was said before. So the second sentence is a result of the first sentence.
examples could be:
My head hurt SO I went to the doctor
I was hungry SO I ate an apple
If you want to check if you have used “SO” in the correct way, you could try and reverse the sentence and use “WHY?”
so applying it to the 3 sentences used…
“I went to the doctor” “Why?” “My head hurt”
“I ate an apple” “Why?” “I was hungry”
“My cell phone needed to be repaired” “Why?” “It was not working properly”
C) If my cell phone is not working properly by tomorrow, I will need to get it repaired.
the word BY can mean “not later than”. it suggests a time limit. It emphasises that the day after tomorrow is too long to wait.
examples could be:
If I do not hear from my friend BY Tuesday, I will call him
If my son does not clean his room BY dinner time, he will not get dessert
“it will need to be repaired” is not actually wrong, I think the phrase I’ve supplied is more how an English person would phrase it.
a and b are both comma splices. You have two independent clauses joined (incorrectly) with a comma. Either insert of conjunction, like the “so” suggested by Benjamin, or use a period and make them two sentences.