Ideal fridge temperature accounting for temperature spikes

The fridge itself does storing, cooling, and dehydrating. But the last part is rather slow, you don't see the effect all that much. It is more prominent at low temperatures (manifests as freezer burn). But try leaving unwrapped bacon slices in the regular fridge and you'll see what I mean. But anyway, it isn't all that interesting what the fridge does, because we only use it for cooling. The interesting things happen in whatever you put into the fridge.

The explanation will be somewhat long even without details. The dishes you described have different reasons to need cooling, so I have to deal with them separately.

First, egg whites. Stiffly beaten egg whites tend to weep. Cooling them slows the process. You know, less Brownian motion, etc. Still, overbeating them is never a good idea :) In this case, there are no fat particles responsible for the foam stability; in fact, the smallest amount of fat can disrupt the building of egg white foam.

The second class are the dishes which contain lots of saturated fat. They are made with (raw) cream, chocolate, butter, or other such stuff. Saturated fat is an amorphous solid, it has no sharp melting point. It gets very soft before it melts, butter is a good example. So these things are kept at fridge temperatures. If they are left to warm again to room temperature, they become much softer than in the fridge. (The exception which gets softer than in the fridge, but not all the way to the state before cooling, is ganache. This is because cocoa contains starch, which thickens. See the part about gelling below).

Third, we have custards. Panna cotta is a good example. They set by way of denaturing egg proteins. This happens at around 85°C, on the stove. They are put into the fridge to get colder, because they taste better that way. They don't get firmer in there, and they won't melt if taken out of the fridge.

Then we have the hydrocolloids, or gelling agents. While technically starch gels in a different way than gums, I'll put them together here. The important point is to know if they are thermoreversible. Starch isn't thermoreversible. If you create a blanchmange thick enough to stay on its own out of the mold, it will need to be cooled before it is set completely. But once set, it won't go fluid again at high temperatures. Gelatin, on the other hand, is thermoreversible. It needs to be heated before it will dissolve at all, but then it must be cooled to set again. If you make jello and keep it outside of the fridge, it won't be much softer than in the fridge. But if you heat it up all the way to 50°C, it will melt to liquid again. Then there are the even funnier ones which will harden at high temperatures and soften at low temperatures, etc. But mainly, you don't have to keep hydrocolloid gels cooled once they are set. In fact, they don't even need fridge temperatures to set. Gelatine is perfectly happy to gel at summer room temperatures. But the cooling takes much longer because of the smaller heat difference between liquid and air. So it sets quick in the fridge, but has to be left overnight in order to work at room temp.

As for the practical meaning of all this: you can keep most stuff outside without a texture problem. The exception of this rule is anything made with saturated fat: cream, chocolate, pastry dough, buttercream frosting. But it is common knowledge that these should be kept cool anyway.

My guess - and this is just a (somewhat educated) guess - is that it's just to promote more even cooking.

Since you're talking about a layered dish, some parts are definitely going to cook faster than others. If the entire dish starts from room temperature, as opposed to fridge temperature, then that means less time is required to cook it through. Less time and less heat required to cook means that all of the layers will be more likely to end up at similar internal temperatures - as opposed to having burned bread, liquefied cheese, or rubbery eggs (I'm not sure offhand which cooks the fastest).

Even if it's not an issue with the thermal capacities of your individual ingredients, you're also layering these each several times over, creating a very dense product, so there would still be a significant risk of the middle layers being undercooked, or the outer layers being overcooked.

You might be able to bake it straight out of the fridge; however, you would definitely have to increase the cooking time to account for the temperature difference, and there are a lot of variables that come into play which would affect how evenly it cooks: the intensity and location of your oven's heat source(s), the density of the casserole, the kind of baking dish you use - I probably wouldn't chance it, at least not when preparing this for other people.

You tend to see the same recommendation for anything particularly dense, such as a roast, or anything layered, such as a lasagna, and generally, you do want to follow those recommendations for the same reason. They cook rather poorly if you cook them from cold or frozen, leaving you with a charred surface and an only-mostly-cooked interior. It can still happen even if you start off at room temperature, but it's less likely and the effect tends to be less pronounced.