"He then" vs "Then He" vs "Then, He" -- conjunctive adverbs, semicolons, and commas

As far as I understand, you use a semi-colon to separate main clauses joined by conjunctive adverbs (however, therefore, moreover, nevertheless, then, thus). And, when you use a conjunctive adverb, put a semicolon (;) before it and a comma (,) after it.

Example, using however:

Ernest Hemingway was a master of style; however, opinions about his work vary widely.

BUT using then, referring to sequencing, what is the proper usage?

Example:

He went to the store. Then he went home.

The above explanation would say that I need a semicolon between the sentences and a comma after then. And I have found many examples of this.

He went to the store; then, he went home.

But, is it a fast rule when you use then (I've read that "then" may not always be a conjunctive adverb) or can you just use a period? Do you always need the comma after, and can you reverse it?

He went to the store; then, he went home.

He went to the store; then he went home.

He went to the store; he then went home. (no comma after "then")

He went to the store. he then went home. (no semicolon between sentences)

I generally use He then as I feel it sounds better, but I'm confused as to whether that is grammatically correct and how to properly punctuate it. I can't find a definitive examples sentences.

You can't make a simple rule about using comma after "then."

When "then" is the first word in the sentence, it acts as an introductory element, after which you usually use coma, but not necessary.

He went to the store. Then, he went home.

If you omit the comma, the sentence is still correct, but the pacing is different:

He went to the store. Then he went home.

You can also say:

He went to the store and then he went home. (no comma)

or

He went to the store; then he went home. (no comma)

You can use "then he went" or "he then went," depending on what is more appropriate in a given sentence.

The Chicago Manual of Style

Q. Is it necessary to use a comma after words like next, then, after that, last, and finally when they are the beginning of a sentence?

A. Punctuation is not so simple that you can make a rule that a comma “always” follows a given word or phrase. Commas depend on syntax as well as pacing, tone, and personal preference. Two examples with next:

Next comes the scene where he buries the toenails.

Next, since he was still breathing, she worked a crossword puzzle.

Do you always need to use a semicolon before "then" and comma after it, because it is a conjunctive adverb?

No. Check this article about Commas With Conjunctive Adverbs (American Journal Express)

You can see that conjunctive adverbs can be used at the beginning of the sentences as "introductory elements," which are usually (but not necessary) followed by comma.