> First of all, "catch fire" is at least partly understandable from its constituent parts - "catch" has a great variety of related meanings, and they all have to do with something taking hold of something else
If you want to analyze it that way, you'll find that the semantics are the reverse of what you predict: when you catch fire, it's the fire that takes hold of you.
> First, the article can very well be pronounced independently - I can say "English has a single indefinite article, with two forms: 'a' or 'an'".
This argument is predicated on forgetting the difference between use and mention. What part of speech would you say an is in that sentence? Is it an article?
> Secondly, the 'a' form can be pronounced in two different ways, depending on how you want to highlight it within a sentence
Yes, problems arise when you need to place sentence-level stress on a feature that is too weak to bear stress. The same problem occurs for any English clitic, including 's, which in the general case doesn't even include a vowel. Most notably here, there's nothing about this specific to a before consonants; the rules for placing stress don't know what word you're following a with. If you need to stress an, the usual choice is /æ/. But also notably, when native speakers do this, they recognize it as a problem - it's just one they may not be able to work around.
> Finally, in at least some English accents, many words can be pronounced differently in certain sequences than others - for example, in modern Southern English, an "r" sound is introduced almost always in speech when a word that ends in a vowel sound is followed by another word that starts with a vowel sound, e.g. "I saw-R-it". By your description, neither "saw" nor "it" are individual word phonologically
This is not a word-level phenomenon in any way; intrusive R also occurs between syllables of a single word, as long as there's an appropriate vowel-vowel sequence. Placing one between saw and it would not normally be viewed as altering the pronunciation of either word (Which one do you think is altered? I guess by nonrhotic standards it would have to be it), but as the application of a general rule.
Placing /n/ between a and elephant is not the application of a general rule, it's the application of a rule specific to a.
> Overall, the atomicity of a linguistic construct is highly debatable, even in a particular context.
You're saying that people argue over which items count as words, not that they argue over what it means to count as a word.