Sentence structure is totally independent of its meaning, or the thing it wants to convey. Your confusion greatly roots on the fact that you are so concerned on the actual meaning of the sentence rather than the structure itself.
But let me demonstrate why "because" falls into subordinating conjunction category:
In your example
He passed because he was intelligent.
Note that there is only one line of thought here: The mere fact that he was intelligent made him pass the exam. In other words, you are basically saying that being intelligent is a cause for him passing the test.
On the other hand
He was intelligent therefore he passed the exam
Note that there are two separate states here: first, the fact that he was intelligent; and second, the fact that he passed the exam. What "therefore" did is to combine these two separate facts in order to make an inference.
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
As rightly explained by Brian, "feeling well rested" is a phrase. A phrase is a group of words giving incomplete meaning. Phrases are to be viewd from the point of construction as well as from that of function.In your phrase-- feeling well rested-- 'feeling' is a particple(partly verb + partly adjective).
Your phrase is a participle phrase by construction and adjective phrase by function qualifying the pronoun "he". You can put this phrase in the beginning or at the end and with a"," as proposed.
In the beginning the functioning as adjective becomes all. the more apparent.