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I probably shouldn't write the explanation, because I like your style, but I can answer the question at the end of part 2.

In $(a (add a 1)), you'll notice that this isn't %-. That would look like ($ (add a 1)), except that doesn't actually work because the $ is in the core, so it's not the whole gate.

The irregular syntax a(b c, d e, f g, ...) is the %= twig, so $(a (add a 1)) is %=($ a (add a 1)). You can look up the docs for that, but, in English, that's "produce $, except replace a in its subject with (add a 1)". You can replace as many or as few things in the subject as you want, and this form works for more than just gates. =+(a=[b=1 c=2] a(b 0)) gives [b=0 c=2], for example.