I found an example from an online dictionary as follows.
彼は私の手伝いをするのを露骨にいやがった。
He showed a great reluctance to help me.
Because の in the relative clause「私の手伝いをする」 can be replaced by が, so
私が手伝いをするの
can mean "the help I do".
Can 私の手伝いをするの mean "the help I do"?
In other words, can the original sentence
彼は私の手伝いをするのを露骨にいやがった。
be translated as follows?
He showed a great reluctance to receive my help.
"Can 私の手伝{てつだ}いをするの" mean "the help I do"?
No, it cannot regardless of the context.
「私（の/が）する手伝い」 can.
"Because の in the relative clause「私の手伝いをする」 can be replaced by が..."
But 「私の手伝いをするの」 is not a relative clause in the first place; It is only a nominalized verb phrase. Therefore, whatever works in relative clauses is irrelevant here.
Let us take a look at a real relative clause here. How about:
「私が手伝っているお店｛みせ｝」 ("the store I help out at")
Since it is a 100% genuine relative clause, that 「が」 can correctly be replaced by「の」.
In other words, can the original sentence
「彼は私の手伝いをするのを露骨{ろこつ}にいやがった。」
be translated as follows?
"He showed a great reluctance to receive my help."
No, that would be an incorrect translation. The correct Japanese sentence that would mean that would be:
「彼は私が手伝うのを露骨にいやがった。」
One exception of so-called "ga-no conversion" is that you cannot use の in place of が if there is を. Such ～の will be taken not as the subject but as a modifier of the object marked with を. According to this answer,
But you can't replace が with の if there's a direct object marked with を:
ジョンが本を買った店
*ジョンの本を買った店 (ungrammatical)
The store where John bought the book
As mentioned in the comment in the link, ジョンの本を買った店 will be taken as "the store where someone bought John's book".
私の手伝い can mean not only "my help" but also "help for me" depending on the context.
Therefore:
1 ≠ 2 = 3 = 4.
No, it can't. You can't replace が with の in your first example without changing the meaning of the sentence.
That is to say, 私の手伝いをするの means "to help me" and 私が手伝いをするの means " the help I do".
