In your example, the em dashes are used in pairs to set off parenthetical phrases. A parenthetical phrase is a clause that is inserted into the flow of an otherwise complete sentence as an "interruption" that adds additional information. If the parenthetical phrase is removed, the remaining words should form a full and complete sentence on their own.
Despite the name, parenthetical phrases do not require parentheses; they can be formed with em dashes and commas as well, as you correctly state. However, your first example is complex enough that I would probably use parentheses to set off the two parenthetical phrases, to avoid a pileup of commas that would make the sentence harder to understand:
Dean London (a business unit of Centrell) have been supplying permeable products to Caramba Unit (a supplier of biological products) since 2009.
Your second example is fine as is, although commas might improve the flow slightly. Em dashes are often thought to apply more "braking action" to the sentence flow, whereas commas provide a gentler "slowing" action:
This initiative, led by Dr. Reiv Nadar and integrated into Dean's research and design unit, intends to gather post-harvest disease control information.
If you do wish to use an em dash, it should not have spaces on either side—use it like this. In some contexts, you can use an en dash or hyphen instead, with spaces on both sides – like this.
The comma is a choice of style. Some style books say it depends on context, others recommend one option or the other.
Those prepositions are simply prepositions. They aren't conjunctions in any way. Whether or not you can move a phrase is not directly related to whether it is a clause or not. I have to admit I don't understand your explanation of why you think they should be conjunctions. Many kinds of phrases are separated from the rest of the sentence by one or two commas without being clauses.
A clause is a finite verb and all its arguments, i.e. a main verb and everything that depends on it. If you have a sentence with a single main verb, the entire sentence is one clause. Conjunctions and relative pronouns typically introduce new clauses.
In the Odyssey, Odysseus poked out the eye of the Cyclops.
The sentence above has a single clause.
The Cyclops whose eye he poked out was named Polyphemus.
Here are two clauses: the entire sentence is the main clause; part of the main clause is the subordinate whose... clause. Each clause has its own finite verb: poked and was. Here whose is a relative pronoun, which always introduces a subordinate clause. This relative clause is defining/restrictive, in which case it is not marked off by commas.
Subordinate clauses are technically part of a main clause, but, in practice, when one says "the main clause", one often means "the main clause excluding its subordinate clauses". Note that some linguists consider any verb to be the core of a clause, not just finite verbs; but I will not do so here.
He maimed the Cyclops but didn't kill him.
Here you have two main clauses. Notice that they can't be moved around, which is usually the case with two coördinated clauses. But is a coördinating conjunction, which means it introduces a clause at the same level, in this case a second main clause.
Troy he no longer thought of.
He no longer thought of Troy.
The object Troy can be moved around. The result is a change of focus. It is obviously not a clause.
Solemnly Eurycleia washed Odysseus's feet.
Eurycleia washed Odysseus's feet solemnly.
The adverb, which isn't a clause, can be moved around.
Best Answer
The teacher is mistaken, though they are equivalent in your first two sample sentences.
"I can't see you -- are you here?" is grammatical, but "I can't see you, are you here?" is a comma splice.
"What the --" isn't a complete phrase, but it's acceptable in dialog. "What the," would not be. Similarly, "What the -- oh, there you are" would be acceptable in dialog, but "What the, oh, there you are" would be a mess.
The last sentence, "My friend–Alex, ran to the store," is indeed incorrect. You can set something off with em dashes or commas here, but not one of each.