I am writing to ask for a three-day extension as my laptop spontaneously combusted last night.

"As" introduces the adverb clause of reason. You cannot use "because of" or "due to" here because the reason is a complete clause ("my laptop spontaneously combusted"), not a simple noun phrase.

Because I lost all my files in the small fire, I have to start the essay completely from scratch.

"Because" is the correct conjunction to introduce a full dependent clause.

I hope you can grant me this extra time, since this was a completely unforeseeable disaster!

"Since" is used here to introduce a reason that is already somewhat understood or logically follows from the previous context.