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What is the difference between the genes identifies as "extra-species sourced" and those which are a nominal part of homo sapiens DNA? How do we know they don't originate from intra-species mutation, for example?

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Genes tend to to accumulate mutations at a constant rate (lots of caveats here, but that's the general idea). If you find that a population has a family of gene variants that are very different from the same ones in another population, then it's a good bet it arrived from external breeding. And the relative differences between the modern descendant genes can tell you when it happened.

Again, I'm sure the real models get really complicated and are no doubt still a little controversial. But that's the basic idea.

How do they differentiate multiple parallel lines lines of mutations coming back together from a separate subspecies existing for much longer?

Genes don't "come back together". If you have a given gene, it's a direct descendant from exactly one of your ancestors in each generation.

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