Can you explain the difference between "complex" and a "compound" sentences to a mathematician?

As a mathematician, I would feel very, very suspicious about an explanation made to “appeal to a mathematical mind”. Especially since I know that English is complicated and trying to press it into simple rules is going to fail.

It seems to me as if a dependent clause is simply an independent clause and conjunction lumped together.

Be careful. This may be true in some cases, but not in all cases.

Now, the distinction between a complex and a compound sentence lies with the conjunction. A compound sentence uses a coordinating conjunction, because the two clauses - both independent - are of equal importance. A complex sentence uses a subordinating conjunction, because the two clauses are of different importance: one is independent and the other is dependent.

Most of your sentences are compound sentences. They can't be changed into complex sentences by "drawing the boundaries" wherever you like. This is because they utilize coordinating conjunctions, and this doesn't change by including or excluding them.

If a computer were tasked with differentiating between compound sentences and complex sentences, searching for conjunctions would be a good way to start (the other way would be to check for independent/dependent clauses, which I think requires intuition that a computer cannot possess).