For example, is it better to just dive into existing code in the language and play around with it, as opposed to picking up a book on the language and first understanding the theory and the idioms behind it end-to-end?
A crucial, well understood advantage with PLs is that they are well documented (both officially and on Stack Overflow), making minor successes with little previous exposure more likely, which can provide constant reinforcement.
In fact, natural languages are far better documented than people believe. Armed with a few electronic unabridged dictionaries, translation corpora, and conjugation tools, you can make text chatting with a native speaker motivating in the same way as diving into a new programming language, even without having ever composed a sentence before.
I doubt it. Even programming languages with very large 'vocabularies' are tiny compared to natural languages and the grammatical rules of any natural language are much more complex than the syntax of any programming language.
Realistically, when you learn a new PL, how much time goes into learning syntax versus other things?