For the second example, the only possibility is:

Cette école a le plus de garçons et de filles.

In this kind of sentence le plus is fixed and refers to the action of having, or being, rather than to the objects it has the most of.

That’s simple, just use le plus de in both sentences. Le plus de is an invariable set expression.

TLFi

2. Le plus de. [Suivi d’un subst. sing. ou plur.] La quantité maximale de, le nombre maximal de. Grande discussion sur Dieu, un des sujets qui font dire le plus de bêtises aux gens d’esprit (Goncourt, Journal, 1864, p.108). On a beaucoup plus respecté la vie humaine aux époques où elle a réellement le plus de valeur (Renan, Avenir sc., 1890, p.499).

Here are other quotes showing le plus de is not informal and still standard French even without grand nombre inserted:

• Le plus sot homme, comme le plus sot peuple, c’est celui qui dit et qui fait le plus de sottises Voltaire.

• Il est attribué à la liste qui a obtenu le plus de voix un nombre de sièges égal à la moitié du nombre des sièges à pourvoir… Art. L262 du code électoral.

• Aussi, lorsque nous nous séparâmes des gardes, M. Deviolaine indiqua-t-il une chasse pour le dimanche suivant, avec ordre de détourner d’ici là le plus de sangliers que l’on pourrait, afin que, si l’on faisait buisson creux sur une garderie… A. Dumas

Good first question. 🙂

TL;DR:

1) de separates le plus and musées; musées is not really the noun introduced by le, even if it looks like it. As a result, le is probably best described as not agreeing with any other word.

2) As the other answerers wrote, le plus de is always correct. However, as Lambie writes below, it may be informal compared to fuller formulations such as le plus grand nombre de.

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Difficulties with agreement are always going to involve the question: “Which elements are agreeing?” This question comes up whether you’re talking about conjugating verbs, modifying adjectives and participles, choosing determiners…

Here, you would begin by asking about le. Which word is it introducing?

The English translation you gave suggests that museums would be the nearest noun:

…the most museums.

But while this is a good translation, it’s not quite the grammar of the French:

…le plus de musées.

This structure, given away by the presence of de, is a common one that usually links two nouns:

Une boîte de farine

Une tasse de thé

Or a quantity and a noun:

Beaucoup de farine

Plus de musées

And in both cases note that the noun itself loses its own determiner:

De la farine

but Une boîte de farine

but Beaucoup de farine

This suggests that in your textbook’s sentence, the determiner for musées would have looked something like this:

Des (de + les) musées

but Plus de musées

All this is to show that the le is not for musées. Its les disappears in this structure.

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So what is the le for in this phrase? I guess there are two ways the analysis could go.

Maybe you could say that plus is short for plus grand nombre, for example, in which case le agrees with nombre (and we would have a noun + noun nombre de musées instead of quantity + noun plus de musées).

But that explanation is a bit too ad hoc for my taste. We could also do it grammatically and start with the rule that the comparative is plus, and the superlative is made by adding the definite article:

Elle est intéressante

Elle est plus intéressante

Elle est la plus intéressante

When we’re comparing an amount of something, we still use plus:

Paris a plus de musées

And we still make it superlative with the definite article:

Paris a le plus de musées

But we notice two interesting things…

First, that neither with comparing adjectives nor with comparing quantities is this le really a normal determiner. After all, even in “la plus intéressante” or any other comparative phrase with an adjective, the determiner is not followed by a noun as you would expect it to be. Probably the determiner just picks up the gender and number features of the next item in the same constituent or “block” of the sentence.

The second (key!) thing we notice is that these are not really the same plus. One is followed by an adjective, and the other by de. Even without further analysis of what they are, that tells us they behave differently.

And if we skip ahead a bit in linguistics we find that de begins a new constituent, the prepositional phrase. That’s enough to separate le from the noun, meaning le won’t have an immediate neighbour from which it would borrow gender and number features. So it retains the unmarked form, le.

There is something LURKING behind this question, which, I had to learn as a non-native speaker, and also, as native speaker of English:

Paris est la ville française avec le plus de musées.
The full and standard way to say it is:
Paris est la ville française avec le plus [grand nombre] de musées.

So, now: Your sentence below:
Cette école a le plus de garçons et de filles.

would become:

Cette école a le plus [grand nombre] de garçons et de filles.

And the English, in standard grammar is then:
Paris is the city with the highest or greatest number of museums.

The school has the highest number of boys and girls.

Commonly expressed as: the most museums and the most boys and girls. Just like French.

Though you commonly would translate these: the most museums and the most boys and girls. Lurking behind the English is: the highest number of, just like French.

Now, if you are writing an academic paper or a formal article, the most in English does not work: you would have to say: the highest or greatest number of A and B.

The most coffee or butter but the greatest number of or the highest number of [countable noun].