Learn Hardcore Japanese: negai ga kanau hi wo matte imasu. - I'm waiting for the day when my wish comes true.

かなう means “to be fulfilled,” “to come true,” or “to be realized.” It is an intransitive verb, so the thing that comes true becomes its subject. In this sentence, 願い is marked with が because it is the subject of かなう.

Why is が used after 願い in 願いがかなう?

The particle が marks the subject of an intransitive verb. Since かなう is intransitive (“the wish comes true”), 願い must take が rather than を (which would make it a direct object) or は (which would make it a topic).

What kind of clause is 願いがかなう日 in this sentence?

This is a relative clause: 願いがかなう modifies 日. Together they form “the day when the wish comes true.” In English: “the day [that] my wish comes true.”

Why is 日 marked with を in 日を待っています instead of に?

Here 待つ is a transitive verb meaning “to wait for (something).” The thing you wait for—日—is its direct object, so it takes を. If you used に you would be making 日 into a time expression (“on that day”), changing the meaning to “I am waiting while being on that day,” which isn’t what we want.

What is the function of the て-form in 待っています?

The て-form of 待つ connects to いる to form the continuous aspect. 待っている literally means “is waiting” (ongoing action or state). Without the て-form, you would just have the simple verb 待つ or polite 待ちます, which don’t emphasize the ongoing nature.

Why do we use 待っている rather than the simple 待つ or 待ちます?

待ちます (or 待つ) can express a habitual action or a future action (“I will wait”).
待っている emphasizes that the waiting is in progress right now and may continue.
In this sentence, the speaker wants to convey that they are currently—and patiently—waiting for that day.

Why can we omit the subject in 願いがかなう日を待っています? Who is the subject?

Japanese often drops pronouns when they’re clear from context. Here, the implied subject is “I” (the speaker). If you need to specify, you can say 私は願いがかなう日を待っています.

Could we rephrase the sentence using the nominalizer の instead of including 日? For example, 願いがかなうのを待っています. If so, what is the nuance difference?

Yes, 願いがかなうのを待っています is grammatical and means “I’m waiting for my wish to come true.”
Nuance difference:

Using 日 focuses on the day itself when the wish comes true.
Using の makes the entire event (“my wish coming true”) the direct object, without highlighting a specific day.

Alternatively, could we express “waiting until my wish comes true” using まで instead of 日? How does 願いがかなうまで待っています differ from the original?

願いがかなうまで待っています means “I am waiting until my wish comes true.”
This phrasing emphasizes the time span leading up to that moment (まで = “until”).
The original 願いがかなう日を待っています highlights the particular day itself. Both are natural; choose まで if you want to stress the duration, or 日 if you want to point to the specific day.