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)

"Whether," means the same thing as 'if.' The sentence could have read "I don't know if we'll go on an outing tomorrow." Had it said that, it would mean the same thing. "That," is used as a short-hand instead of saying "The fact that....(he finished writing the composition.....)"

These sentences use whether and that instead of a noun. A simple sentence will be: (1) It surprised us all. What surprised us all? That he finished the composition.... (2) It remains unknown. What remains unknown? Whether or not we will go for an outing... The difference between whether and that is that that is certain and whether is not certain. He definitely finished the composition, but we are not sure if we are going for an outing

没有that 不可以么？

That指代整件事情 在surprised us all中作主语 Whether 在那里表示是否 引导主语从句时放句首