No, that's not quite what's happening here. As it is, the preposition "на" (as well as "в") takes the locative case - which, for all intents and purposes, has merged with the prepositional case. So the expected Russian sentence is "На мосте ничего нет".
But that's not the case.
The thing is, in a number of second-declension masculine nouns, the original Proto-Slavic locative form has survived and is still used to this day. Some of these nouns are:
лес - жить в лесу - but говорить о лесе
мост - стоять на мосту - but думать о мосте
снег - в снегу - but мечтать о снеге
рай - жить в раю - but читать о рае
дом - жить в доме (no distinct locative ending except for one set phrase) - but работать на дому.
Note that the locative-case form coincides with the dative, but there's a slight difference in that the stress is shifted to the final syllable:
Я живу в лесу́ - locative.
Я иду к ле́су - dative.
I beg you guys, please don't use italics with the Russian alphabet. We are just learners, we are not skilled enough to read it easily. Another thing: please, provide translations for your examples! Nonetheless, thank you for the detailed and precise reply.
Would saying "На мосту ничего" be okay, or is it bad Russian? How necessary is the "нет"?
нет is a contraction of "не есть," in short it's the verb in the sentence. Forms of "to be" are normally omitted in the present tense in Russian, but not in sentences where the emphasis is on existence—or, to the best of my knowledge, lack of existence.
Is there not a need here for это? It seems the direct translation here would be, "on the bridge nothing", which is not a sentence. How can we know in forming Russian sentences that the implication is enough?
I don't think это would be likely to make an appearance in a translation of an English sentence like this; it would be a word by word translation that doesn't work in Russian.
The verb here is "нет," a contraction of "не есть," so it means basically "is not" or "does not exist."
So the direct translation here is better rendered "on the bridge nothing is not," which yields the suggested English version when accounting for the different meanings of double negations in Russian and standard English.