Let's tackle your first example first (I've changed the name of Hunting Lodge to set aside your notion that "the Hunting Lodge" is a common noun phrase):
Casa Loma consists of three main buildings — the Casa Loma proper, which is a seven-story Gothic tower, the two-story Cabot Vicorage that once housed the home’s servants quarters, and a massive stable.
Let's ignore those comments on your previous question for a moment and ask ourselves which is more appropriate – that or which?
I'm going to quote from a Writer's Digest grammar blog to get one opinion:
The battle over whether to use which or that is one many people struggle to get right. It’s a popular grammar question and most folks want a quick rule of thumb so they can get it right. Here it is:
If the sentence doesn’t need the clause that the word in question is connecting, use which. If it does, use that.
The blog continues with an example, and explains:
Our office, which has two lunchrooms, is located in Cincinnati.
Our office that has two lunchrooms is located in Cincinnati.
These sentences are not the same. The first sentence tells us that you have just one office, and it’s located in Cincinnati. The clause which has two lunchrooms gives us additional information, but it doesn’t change the meaning of the sentence. Remove the clause and the location of our one office would still be clear: Our office is located in Cincinnati.
The second sentence suggests that we have multiple offices, but the office with two lunchrooms is located in Cincinnati. The phrase that has two lunchrooms is known as a restrictive clause because another part of the sentence (our office) depends on it. You can’t remove that clause without changing the meaning of the sentence.
My guess is that you read similar advice somewhere, and that's what prompted you to write your original question.
In your example, let's assume there is only one Cabot Vicorage (or only one Hunting Lodge), so, if we follow the published Writer's Digest advice, our choice should be which, because we can remove that part about the servant's quarters and not change the meaning of the sentence:
Casa Loma consists of three main buildings — the Casa Loma proper, which is a seven-story Gothic tower, the two-story Cabot Vicorage which once housed the home’s servants quarters, and a massive stable.
So, why are the people on ELL and on the Word Reference forum not immediately saying so?
I think the key part of the blog I've quoted is the opening sentence:
The battle over whether to use which or that is one many people struggle to get right.
In other words, most native speakers don't even think about whether or not they are dealing with a "restrictive clause" when they are talking. If they majored in English, a thought like that might momentarily cross their mind while they are composing an email and wondering if they should use that or which.
However, you keep leaving out a lot of key information. Unless prompted, you don't say where the sentence came from. You don't cite any English grammar rules. You don't mention what level of correctness you are striving for. You don't clarify anything about the Hunting Lodge. You just throw out some cryptic sentence and expect everyone to know which word should be used, so they start analyzing it by what "sounds" acceptable, and don't research it any further.
With that all said, the sentence in your second question should follow the lunchrooms example, using two commas along with the word which:
Why did the United States, which had welcomed so many millions of immigrants for nearly a century, suddenly become so fearful of immigration in the 1920s?