As the post states, “If the clauses are short (your call), then leave out the comma.” In our opinion, “he loves jazz” is short enough to allow for inclusion or omission of the comma separating the clauses according to a writer’s style and preference.

It is my understanding that “he loves jazz” is an independent clause; therefore, it can function on its own. If it can function on its own, then “and” is a coordinating conjunction thus requiring a comma. Thoughts?

In reference to Genesis 1:27, the text that appears in biblical translations such as the New International Version appear to favor literary technique over strict grammar; the style is for sound and effect rather than precision. Other biblical versions appear to have applied the editor’s pen, such as this treatment in the New Living Translation: “So God created human beings in his own image. In the image of God he created them; male and female he created them.”

Your sentences are punctuated correctly; no additional commas are needed.

So God created man in his own image, in the image of God he created him; male and female he created them.

Since commas alone aren’t able to join two independent clauses together, I assume everything up until the semicolon is regarded as one main clause, and that which follows after it is regarded as another; therefore, making this example of mine a compound sentence. But I’m unsure due to the comma. If this is indeed a compound sentence what is the function of the comma?

I look forward to your response.

Examples:
And Elaine Piper, who played many roles, was a formidable character in real life.
But many of the students scored well on the tests.
But for whomever you are contacting, make sure you use the correct form of address.
So when you need help, please call me.

As this post states, “Use the semicolon if you have two independent clauses you are connecting without a conjunction.” Your second example is correct, or you may write it as follows:
Additional testing may be performed; please contact the laboratory if further testing is needed.

In accordance with our Rule 3 of Semicolons, “Use a semicolon to separate units of a series when one or more of the units contain commas,” your treatment would help make the sentence easy to read and understand.