Romaji is a curse. It creates bad habits, and slows you down in the long run. Avoid using Romaji at all costs if you ever hope to be proficient at Japanese. If you don't believe me, look up the studies saying so. It's not a debate, it's something that's generally agreed upon by experts
Not really. Article is not always required for it to be a noun. Articles are just determiners, like adjectives. What distinguishes verbs from nouns is how they're used in a sentence (position, context etc.). Without that context, one can't really tell. And that's the whole problem: The question didn't provide context, so the learner is none the wiser.
のむ is either to drink as the dictionary form or as a predicate in the casual form "drink" (as a verb). のみます is the polite form of のむ if it's used as a predicate, but it can not really be translated as "to drink". And as mentioned in other comments before, のみもの can indeed be translated as beverage or drink (as in the noun).
Less than many other languages, actually. For one thing, Japanese doesn't inflect for person or number. For another, it doesn't actually have past/present/future, but rather marks for completed/uncompleted. And much of 敬語 (けいご) - polite language - is a matter of different verbs, not conjugation.
Desu is a copula similar to "is/am/are" used when making "A = B" statements and wouldn't make much sense here.
watashi wa nomu works fine for "I (will) drink" or watashi wa nomimasu if you want to be a bit more polite. Though the "watashi wa" part entirely can be cut out if it is already understood from context that you're talking about yourself, so a simple "Nomu" or "Nomimasu" for "I drink" would be a perfectly fine complete sentence in Japanese.
There are formal and informal forms of all verbs, yes. This here is the informal present/future form のむ that you would find in a dictionary. Then there is the polite present/future "-masu" conjugation of the verb のみます. This is the form that many language programs start with since its more formal.
Then there's the -te form conjugation のんで that is used for commands "drink!" and then the progressive -ing form of the verb "drinking" would be のんでいる(informal) or のんでいます (formal) that I have also seen many language programs start with (which I think just makes it way more confusing). Don't worry about these forms yet; they'll be covered later when you've already got the hang of the present/future.
osake/sake (they're the same, just the first is more honorific/polite) is indeed alcohol, a type of drink (noun). "a drink"/"drinking" in English can often be used casually to refer to alcoholic drinks, and is probably why that answer is acceptable.
Nomu is the literal verb "to drink" - the action of drinking something
You drink (のむ - verb ) a(n) (alcoholic) drink ( おさけ - noun )
"Nomu" is the unconjugated form of the verb, also used as the casual present/future tense. This is the form you'll see in dictionaries and often used in casual speech.
"nomi" is just a verb stem and you won't see it used by itself. It is used to build polite form conjugations like the present/future "nomimasu", negative "nomimasen" and past "nomimashita"
You are correct that all verbs in their dictionary form will end in an "u" sound, though Japanese doesn't really have an infinitive. The unconjugated dictionary form is used as the casual non-past tense. Polite non-past will be conjugated into -masu form のむ nomu (casual "drink/will drink") becomes のみます nomimasu (polite "drink/will drink")
Duo mostly focuses on polite speech early on, so you'll see a lot of verbs ending in -masu or the negative -masen in upcoming lessons which are a bit easier to conjugate, but the dictionary form will be very important for the more complex conjugations like casual forms, the imperative and progressive.
watashi wa osake o nomu
Word order is a bit different in japanese; the verb goes at the end. You also need these things called particles to mark the function of words. "wa" marks yourself as the topic, the one doing the drinking, and "o" to mark osake as the object that you are drinking. :)
飲む・のむ・"nomu" is the verb "drink" in casual/dictionary form. (kanji, hiragana to show pronunciation, romaji pronunciation)
飲みます・のみます・"nomimasu" is the verb "drink" conjugated into its polite form.
飲み物・のみもの・"nomimono" is the noun "drink" made up of the kanji "drink" and "thing". Literally "Drink thing"
All verbs in Japanese end in an う syllable. Depending on the conjugation, the last syllable will change. For ます form of regular verbs, you change the last syllable from an う syllable to an い and add ます. For いる/える verbs you drop the うsyllable and add ます. David already used のむ as an example of regular verb. ます form of いる is います.