The answer to your first question, like almost all "why" questions, is "because that's how English is".
For more explication: in a relative clause one (but not both) of the subordinator "that" and a relative pronoun "which" "who" must be used; except that if the relativised noun phrase (the one to which the relative clause is attached) functions as the direct object of the verb in the relative clause, then they may both be omitted. Note the word "may": "that" or "who/which" may be used or not, at the speakers choice.
These are all relative clauses attached to a noun phrase, so cannot normally occur at the beginning of a sentence.
Your final example is a very different construction. As others have pointed out, it is ungrammatical as it stands; but if the subordinate clause is corrected to "that the weather was so bad", it will be grammatical; but it is not an instance of a relative clause, but a nominal clause, the subject of "was not the fault of the forecaster". For that construction, as you say, "that" is required if the clause begins a sentence, but is usually optional if the clause follows a verb and is its direct object.