1) One might be tempted to say that you yourself slipped a bit there.
2) Both semicolons and colons shall be followed by a capitalised word as what actually follows is a whole new sentence!
1) One might be tempted, but one should resist the temptation. The sentence in which you think I slipped is this one: "In this case, the that that I had to say was a sentence I'd previously written." I take it you believe that the reduplicated
thats are a slip-up on my part. Not so. The sentence to which you object follows the following sentence: "But a colon has the force of the word 'namely,' 'to wit,' or 'that is to say.'" The first
that in the sentence to which you object is picked up from the phrase "that is to say" in the preceding sentence. In the sentence to which you object, I treated that
that as a noun, effectively (something)-that-(is-to-be-said) (the material in parentheses understood from the previous sentence). Then I followed that
that with the relative pronoun
that: the
that of which I am speaking, that
that. I did all of that, of course, precisely to allow the reduplication. You see, I'm on record arguing on behalf of the view that two of the same thing next to one another constitute a great beauty.
Is that clear?
2) Not ordinarily, no. In the case of semicolons, a capital letter should follow only in instances where a proper noun begins the second independent clause. Semicolons don't link two sentences; they combine two independent clauses into a sentence. An independent clause is a unit of grammar; a sentence is a unit of orthography (defined precisely by beginning with a capital letter and ending with a period). In English orthography, two independent clauses may be written as two sentences, each beginning with a capital letter and ending with a period, or as a single sentence where the independent clauses are separated by a semicolon, but the first word of the second independent clause is not capitalized (unless, again, it is a proper noun).
In the case of colons, only in rare instances like my own, where the thing generally described in the independent clause preceding the colon, and needing greater specification in the (usually, but not here) noun phrase after the colon, is itself a sentence.