Italics are quotes from other people (someone different in each case); plain text is me.
Is cereal a soup? Is a hotdog a sandwich?
No and no.
No to the first because cereal is prototypically a cold sweet breakfast food, while soup is a warm savory dinner food.
And no to the second because sandwiches are prototypically characterized by horizontal layering of fillings between slices of bread, while the hotdog is self-contained and nestles vertically within a bun.
But what about GAZPACHO?
Gazpacho is cold. So is vichyssoise. But they're both still savory and traditionally served for dinner. Plus, they're vegetable-based rather than grain-based, and prepared by a cook not poured from a box, and so on and so forth. Thus, despite the atypical temperature, they still possess more characteristics matching the mental prototype for "soup" than they do for "cereal" and are categorized accordingly.
Similarly, although the hamburger has horizontal structure, it's a self-contained meat unit in a bun, rather than layers of filling between bread, and therefore it feels like it doesn't quite match the mental prototype for "sandwich". It has a lot in common with the hot dog, but the shape is all wrong, so it doesn't match well there, either. If we were from a different culture where these weren't common foods with strong prototypes, we'd probably be comfortable lumping it in with one or the other, but since we have a strong sense of the hamburger as being a specific thing, we end up forming a new prototype for "burger".
Which works beautifully until somebody swaps out the ground beef patty for a grilled chicken breast and you have to decide what to call something that falls perfectly into the overlap between "sandwich" and "burger"...
Except for all the signs from the 1920's forward advertising for a "hamburger sandwich"
Yes, and that was when hamburgers were new and our culture didn't yet have a sense of them as being a distinct thing warranting a prototype of its own. The past is a foreign country, after all.
The usage of "hamburger sandwich" starts dropping after 1940, while just plain "hamburger" continues to grow. It's overtaken by "burger" around 1970, which I think is indicative of the development of a new prototype that can accommodate variations like "turkey burger" and "veggie burger", both of which start showing up with some frequency not long after.
I don't think sweetness or temperature should be used as a deciding factor in whether something is a soup. The important quality of soup is that it is mostly a liquid essence (french onion, tomato, chicken, etc with broth) that may or may not have chunks. Milk is not an essence and can't be used to make soup unless combined with an essence. I don't know much about gazpacho but it is majority liquid essence. Cereal, oatmeal, porridge, aren't soups, but they are a category very similar that uses milk instead of an essence.
Sandwiches are a free-standing, not-stuck filling contained in bread for easy handling. Hot dogs are sandwiches. Hamburgers are sandwiches. They are special classes of sandwiches that are never referred to as such. Two slices of pizza is not a sandwich, tacos don't use bread. Open-face sandwiches, wraps, and other bread conformations contain the free-standing filling in bread. If I hollow out bread and fill it with something not a liquid essence (like porridge), that's a porridge sandwich.
I hold that there is no deciding factor. The way that our brains work is that we perceive objects as having a collection of associated features, some more central than others. We decide whether an object is a member of some abstract class by comparing its features to the feature cloud of an imaginary mental prototype; if the barycenter is close enough to the prototype, we say, yeah, that's an X.
For soup, "liquid" is one of the most central terms, so it's hard to do without it. Unless you have, say, frozen soup. Or dehydrated soup. Which are both still soup! Or you're at a fancy molecular gastronomy restaurant and they turn the soup into foam or gel or bubbles so that it IS soup and yet is also not-soup, both at the same time, whoa, crazy!
Prototype-based categories are fuzzy contextual clouds that rarely have sharp boundaries between them. Often they are nested. A BLT and a veggie burger are both "food", and both would fall within the cloud of "sandwich" if the only other categories on the menu are "salad" and "soup", but if you've also got a section for "burgers", that's clearly where the veggie burger belongs.
If you fold a piece of pizza in half, you have moved it away from the prototype for "pizza" and towards the prototype for "sandwich"; whether that warrants calling it a sandwich is context-dependent. Tacos don't use bread... unless that's all you have in the house and it's Taco Tuesday and it's been a rough day and you promised the kids tacos, in which case those are 7-grain taco shells, dammit. I wouldn't normally call a hollowed-out boule of sourdough filled with artichoke dip a sandwich, but if you want to put the lid back on it and eat it with both hands, sure, that's sufficiently sandwich-y treatment to push it back over the edge.
Prototypes are also subjective. They tend to match up well from person to person within a society, but they vary culturally and regionally and even personally. That's what makes cross-cultural questions (like "is pho porridge?"*) difficult to answer. And sometimes the differences in prototypes can be quite surprising, especially if you weren't expecting there to be any:
Separated By A Common Language: Prototypical Soup
Separated By A Common Language: Sandwiches, Particularly Bacon
[* A question that came up in a thread that branched to the discussion of whether oatmeal was soup or cereal: since pho is a warm and savory breakfast food, does that make it a very wet porridge?]