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This is a form of だ.
Here we have a “A is B” sentence (AがBだ), but rather than ending the thought there, there’s an “and” followed by her next sentence. That change the だ to で (AがBで).
You’ll all see this with verbs, where they get て or で added on to string together multiple actions in one sentence, or to join together two or sentences.
What you’ve determined is correct. It can be “they say …” or “it’s said that …” or even (depending on context) “is known as …” or “is called …”
For this dialogue, I might go with, “This is a famous pudding. They say it’s really delicious.”
Note that the Japanese って alone doesn’t tell who said it, only that it has been said. In English, we tend to fill in a pronoun for an otherwise unstated speaker (“they say”, or “it’s said that”).
This is where having pronouns left out because they’re known from context can make things more difficult for a language learner. Since this is 寂しい rather than 寂しがる, I agree with your translation of “I’m so lonely.”
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It’s a bit hard to tell, but this is actually…not a つ!
I know it looks like one, but I assure you, it is…not!
It’s actually a っ!
There are resources out there that explain this usage (I don’t know any off the top of my head to link here), but you can preliminary think of it as being similar to ! at the end of a sentence.
You need to use a little bit of HTML for that.
The proper HTML is a bit complex:
<ruby><rb>kanji</rb><rp>(</rp><rt>kana</rt><rp>)</rp></ruby>
So some people prefer to use the HTML minimal that does the job:
<ruby><rb>kanji</rb><rt>kana</rt></ruby>
There’s also a script somewhere for use on WaniKani forums to streamline it a bit.