Here are the answers. There will be no Part 3, I must say...
__________

Why is it that both の and が can be used in that sentence but not the following sentence?

snbzk, the way you should think of it is like this:

In an adjective clause (the food that my dad makes), take out the partial sentence (I don't know the correct term) from the clause (that my dad makes). Any が there can become a の. For example:

父が飲んだビール〜
父の飲んだビール〜

母が作った靴下〜
母の作った靴下〜 (anyone appreciate my unintentional reinforcement of gender roles? ;))

I don't know what "following sentence" you're referring to (I'm guessing you forgot to paste a sentence in your post), but that's my explanation of the の／が phenomenon.

Nagoyankee: Would you say that the use of が used to be favored, but as times change, preference for の has grown, and that's why the latter sounds softer than the former?

Because, as the subject marker, の can only be used in subordinate clauses, not in main clauses.

I was talking about the next sentence in the quiz: "これは昨日私が買ったシャツです。" 昨日私が買った is an adjective clause, is it not? And isn't an adjective clause a subordinate clause?

Alright, snbzk, I've finally found the answer after a weekend of research. The explanation in my last post wasn't too accurate, I apologize.

の and が are only interchangeable when they are used in the clause that constitutes the subject of the sentence.

In the sentence 「父（ が/の　）作るラーメンはとてもおいしいです。」, ラーメン is the subject. Therefore, you have a choice of の or が.

But in the sentence 「これは昨日私（　が　）買ったシャツです。」, the subject is これ. Therefore, the only possible particle after 私 will be が.

Nagoyankee,

「これは昨日私（　が　）買ったシャツです。」
I think this sentence can also be use の.

Do you mean このシャツは私が買いました。？

No, I mean the particle after the 私.

In my ear, の would sound okay only if the word 昨日 wasn't there. But even without the 昨日, が still sounds "better" or "more natural". This may merely be a personal preference, though.

Thanks for your efforts, Nagoyankee. While the issue doesn't seem to have been totally resolved, I'll take your advice anyway and only use の in the subject.

Just for the sake of argument, I found the relevant point in the book where I originally heard about the phenomenon:

All right, I agree.
From the point of grammar, 私の買ったシャツ is also OK.
(subject)の（verb/adjective）(noun) and (subject)が（verb/adjective）(noun)are interchangeable.

Let me explain my feeling but I am not sure it is true.
私が買ったシャツ is (私が買った)＋（シャツ）
私の買ったシャツ is (私のシャツ)＋(買ったシャツ)
Therefore,
（このシャツは）私が買った。 is OK.
（このシャツは）私の買った。 is NG.