Here's my two-cents' worth: (an idiom for you)
That he finished writing the composition in such a short time surprised us all.
--This "that" has no meaning. Its function is simply to serve as a visual/audible flag to the reader/listener that a LOT of words are about to follow (i.e., a noun clause), and that ALL these words will be treated as a SINGLE NOUN which will likely be the subject of a verb that will follow AFTER all these words.
Whether we will go for an outing tomorrow remains unknown.
-- This "whether" indicates a choice between two alternatives (we will go/we will not go). It is necessary to the meaning of the sentence. If these choices are a simple yes/no question (Will we go? yes/no), then IF can be substituted for WHETHER.
-- If the choice is NOT a yes/no question, the WHETHER is required and IF cannot be substituted:
Whether we cancel the project or borrow some more money is something we need to discuss at the next meeting.
(alternatives = cancel or borrow)