Placing "If-then" among conjunctions

You already know about subordinating conjunctions like the conditional if. To my mind, there's another if-then that belongs to another category (you mentioned correlative conjunctions).

For your first question, comma usage will vary. Some books say that because the two parallel parts being combined are found in different clauses, the comma becomes necessary:

If it's a square , then it's a rectangle.

I'd say it depends on the length of the actual clause being linked.

For your second question, "then" is used to emphasize the result/ conclusion that it introduces. So placing it at the beginning of the sentence makes it awkward.

You could do it inside a bigger context though, where the conclusion being marked is still clear:

A: It's a square.

B: Then it's a rectangle if it's a square.

For your third question, there are parallel structures being combined in if-then. They are:

If + S + V, then S + V.

Probably because most of the paired conjunctions (both...and, either...or, not only...but also...) are followed by words or shorter phrases, you hesitate to call clauses parallel structures.

But "either...or" can be this:

We'll eat either here or there.

and also this:

Either you're going to eat out, or I'm going to cook.