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Please comment on the following analysis and answer my queries in question #2. Many thanks.

1. Preparing for the speech, John couldn’t help but feel nervous. (the underlined phrase is a participial phrase functioning as an adjective modifying John.)

2. While preparing for the speech, John couldn't help but feel nervous. (the underlined phrase is a gerund phrase functioning as a noun; but what part of speech is this noun? is it object of the preposition ? can the word while function as a preposition? )

DF
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2- It is not a noun; it is an absolute clause. 'While' is a conjunction.
Comments  
Df20061. Preparing for the speech, John couldn’t help but feel nervous. (the underlined phrase is a participial phrase functioning as an adjective modifying John.)
The participial clause "Preparing for the speech" is functioning as an 'adjunct' of the kind commonly called a 'supplement'. Notice how it's set apart from the main clause by a comma. That tells us it's a non-integrated optional element whose role is to provide additional non-essential information about something in the main clause, in this case the subject "John". But it doesn't actually modify John. Also, note that 'adjunct' (not 'adjective'), is the name of its function. 'Adjective' is a part of speech, not a function. The distinction between part of speech and function is important - see below.
Df20062. While preparing for the speech, John couldn't help but feel nervous. (the underlined phrase is a gerund phrase functioning as a noun; but what part of speech is this noun? is it object of the preposition ? can the word while function as a preposition? )
Yes, "while" can be seen as a prep (though most people call it a subordinator), in which case "preparing for lunch" is a gerund clause functioning as its complement. But the clause is not a noun (though its function is noun-like). Function and part of speech are quite different concepts. The clause is functioning as complement of preposition, but its parts of speech are verb ("preparing") + PP ("for the speech") comprising prep ("for") and noun phrase ("the speech) as object of "for".

It's a common mistake to confuse function and part of speech. Just remember that 'noun', 'verb', 'adjective', 'adverb' etc. are parts of speech, whereas functions are 'subject', 'object', 'complement', 'modifier', 'predicate' and so on.

BillJ