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In the refrain or chorus of the Russian national anthem, there is a line that goes: "Славься, Отечество наше свободное."

Word for word, that might read (to me, at least), "Glory to the fatherland, our freedom."

That doesn't quite make sense, even to me. A better translation might be, "Glory to our fatherland, [which is] free."

My confusion arises from the fact that in English, adjectives (almost) always precede nouns. Is this true in Russian or not? That is, can the noun Отечество be followed by one or two adjectives instead of being preceded by them? Is it true that it's the word forms and declensions and not word order that determines the function of each word in a sentence?

Your translation is incorrect, "свободное" is free, an adjective, not a noun, freedom, which is "свобода". In modern Russian the majority of adjectives indeed precedes nouns, so one will say "наше свободное отечество".

In poetry and lyrics, however, it's pretty acceptable to put adjective after noun - for the sake of rhyming and/or metrical foot.

There's also bunch of phrases where adjective comes after the noun (or at least used alongside with "normal" form pretty widely) - like вечер добрый, дубина стоеросовая etc.

Other - extremely small and quite exotic - category of adjectives that usually are put after the noun - is one that contains indeclinable adjectives of foreign origin, like "пиджаки беж". This is more of a curiosity actually.

Finally, I want stop your attention on following fact. Compare two sentences:

Вася, смелый и ловкий, стукнул медведя по носу деревянной ложкой.

and

Смелый и ловкий Вася стукнул медведя по носу деревянной ложкой.

The order in first sentence sort of emphasize the fact that Vasya is brave and deft, while in the second sentence it's rather an established fact.

It is easy. )))))

Just bear in mind, that a noun and its adjective always agree in gender, number and case.

Отечество - neuter, singular, Nominative. свободное - neuter,
singular, Nominative.

Adjectives have endings -ый/ий(m), -ая/яя(f), -ое/ее(n), -ие/ые(pl) in the Nominative case.

They are not nouns, except for several nouns derived from adjectives and used absolutely, like учительская (teachers' room in a school), ординаторская (doctors' room in a hospital), outdated мертвецкая (a room for corpses in a hospital), and several others, they do not change their gender, and they are found in dictionaries as nouns. Earlier they had a noun (for the three named words it was комната).

Flective languages have a relatively free word order, as inversion (when you verb nouns, etc.) is hardly possible with them. We have endings, which define: a part of speech, gender, number, case, person, plurality, tense. Prefices and endings define perfective verbs. One word contains enough grammatical information.

Meanwhile, such languages as English, with neither cases nor genders, have no other possibility to retain logic in a sentence but putting words in a strict order. You recognize relations between words by their order, and we know that from endings.