Use of semicolon

Q1 I'd prefer a dash here (the modern version of a colon) as what follows is almost a restating (a fairly close paraphrase) of what is said before. But a colon (old-fashioned) or semicolon is not incorrect. And two sentences can be used to add gravitas.

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Q2 This is an old-fashioned use of the semicolon. I'd certainly switch to

• Some people felt that he had “gone mad” or that he harbored some “secret sin”, while others felt it was “an eccentric whim.”

or

• Some people felt that he had “gone mad” or that he harbored some “secret sin” – while others felt it was “an eccentric whim.”

But

• Some people felt that he had “gone mad” or that he harbored some “secret sin”. While others felt it was “an eccentric whim.”

shows the use of a sentence fragment (also present if the semicolon is retained); many now accept the judicious use of fragments to add drama.

... These usages can be checked on by searching on ELU for 'semicolon', 'colon', 'dash' and 'fragment'.

The semicolon is used to join two closely related independent clauses. In your second sentence, you have used it to join an independent clause with a dependent one. It is considered incorrect:

*People felt that he had “gone mad” or that he harbored some “secret sin”; while others felt it was “an eccentric whim.” (incorrect)

You need to use a comma instead or remove the subordinating conjunction:

People felt that he had “gone mad” or that he harbored some “secret sin,” while others felt it was “an eccentric whim.”

People felt that he had “gone mad” or that he harbored some “secret sin”; others felt it was “an eccentric whim.”

People felt that he had “gone mad” or that he harbored some “secret sin.” Others felt it was “an eccentric whim.”

Your first sentence has quotation marks in a rather odd place, but that does not affect the use of the semicolon; you do have two independent clauses. It is correct but ungainly:

His presence was always looked forward to on the Sundays when he filled the pulpit, and he was a ready pastor at the weddings and funerals of Milford; he “had the reputation of a good preacher.”

It's ungainly because, actually, you have three independent clauses. The first two are joined by the coordinating conjunction and.

I would end the sentence after the first two clauses and start fresh (and move the quotes):

His presence was always looked forward to on the Sundays when he filled the pulpit, and he was a ready pastor at the weddings and funerals of Milford. He had the reputation of a "good preacher.”

Note that while colons and semicolons go outside the end quote, in American English periods and commas go inside the end quote.

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Further reading: Grammarly: Semicolon

Uh-oh. The semicolon wars.

In your first example, it's simply a matter of style. The writers want the thoughts to be more closely connected than they would be as separate sentences.

It's a device that some people love and others hate. It was done more often a hundred years ago than it is now.

In your second example, I agree: It should be a comma.

(And now you know which side I'm on. I use colons to connect thoughts more closely than they would be as separate sentences. . . .)