Does comma placement before/after a prepositional phrase alter a sentence's synt...

I am looking to understand when a comma, involving a prepositional phrase, shifts from being a mere stylistic choice to a factor that determines the strict grammaticality of a sentence. For standard descriptive endings, a comma feels purely optional or stylistic:

• "We decided to meet at the park, near the old oak tree."
• "We decided to meet at the park near the old oak tree."

However, when we move the phrase to the front or introduce structural ambiguity, the lack of a comma seems to create structures that feel syntactically invalid or fundamentally broken. For example:

• Case 1 (Garden-path parsing): "While eating, the cat scratched at the door." vs. *"While eating the cat scratched at the door."
• Case 2 (Inversion/Subject roles): "Inside, the house was warm." vs. "Inside the house was warm." (where "Inside the house" might be misparsed as the subject of "was").

Is there a formal grammatical rule in English syntax where the absence of a comma before or after a prepositional phrase moves a sentence from "stylistically awkward" to strictly ungrammatical? Or does punctuation exist entirely outside the bounds of syntactic grammaticality?

In this pair, a comma feels purely optional or stylistic:

1. We decided to meet at the park, near the old oak tree.
2. We decided to meet at the park near the old oak tree.

The comma above does more than aid readability: sentence 2 is technically ambiguous, because although it is likely pragmatically understood as sentence 1, its most immediate parse—as "meet at the park that is near the oak tree"—is absolutely legitimate.

In the next two pairs, is the omission of the comma a violation of strict grammaticality or merely stylistically awkward?

3. While eating, the cat scratched at the door.
4. While eating the cat scratched at the door.
5. Inside, the house was warm.
6. Inside the house was warm. ("Inside the house" might be misparsed as the subject of "was").

Your suggested alternative reading of sentence 6 is quite marked, because we'd normally say "It was warm inside the house" rather than "[Inside the house] was warm". Sentence 6 would probably ultimately be understood as the casual or text-messaging form of sentence 5. Likewise for the preceding pair of sentences.

Does punctuation exist entirely outside the bounds of syntactic grammaticality?

Whether punctuation falls under written grammar or belongs strictly to orthography, one of its functions is to distinguish between competing syntactic parses. Even under the former classification, nonstandard punctuation isn't automatically a grammatical defect; sentences 4 and 6 are informal written English, but all six sentences are grammatical.