They are all grammatically correct (as far as I'm aware), but I think they give different feelings.
Parentheses feel like an aside to the audience, like a weaker form of footnotes. I use them a lot in technical writing — often to indicate bits that are optional or additional information/explanation, but that aren't necessary to read — and in more informal writing (like this post) in a similar sense. They tend to be either quite short, or very long (sometimes multiple sentences, in which case they might be better as footnotes). Just don't forget to close the parenthesis at the end — it sticks out if you don't. I don't think they get used very often in fiction writing, possibly because they break the fourth wall and pull the reader out of the story. It's a case of me, the author, speaking directly to you, the reader.
Dashes feel like more substantial interruptions. I think I use them for asides that are less optional (such as the one in the previous paragraph). Sections separated by dashes can hold less than parenthesis, usually no more than a sentence-worth. I've seen people put sentence breaks inside such sections, but it really sticks out (in a bad way). Again, I don't think dashes get used often in fictional writing, though possibly more than parentheses.
Commas (in this context) are the weakest of the three. They can "hold" the least amount of content, but they are also the least noticeable (which is probably why they can contain the fewest words). As such, they don't stand out to the reader — they may not even be noticed consciously — but they do inform the phrasing of the writing.
I don't normally consciously decide between which of the three I use, it's whichever feels right. It also gets a bit more complicated with dashes as they can both be used in other situations, often not as a matching pair (such as the comma in this sentence).
Looking at your three examples, I would say the following:
The first one — dashes — emphasises that it's his third meal of the day, making it at least as important an idea as the broken table.
The second one, commas, downplays the idea, making the broken table more important, just leaving us with an impression about the eating. This is the most subtle of the three.
In the third one (parentheses), the narrator is speaking directly to the reader and is making an out-of-character remark about Bilbo's eating habits. This is the most blatant of the three.
(I would also suggest that he "noticed that the table was broken", but that may just be being picky or a US/UK difference. And I also apologise if this post has rather too many commas, dashes and parentheses.)
