Learn Hardcore Norwegian: I løpet av kvelden drikker jeg bare vann.

I løpet av is a very common fixed phrase meaning during / in the course of / over the course of (a period of time).
You normally treat it as one unit: i løpet av + a time period (e.g., i løpet av dagen, i løpet av uka, i løpet av kvelden).

Why is it kvelden (definite form) and not en kveld?

Kvelden is the definite form (the evening). Here it suggests a specific evening that’s relevant in context (e.g., this evening / tonight / that evening).
You can say i løpet av en kveld, but that would sound more like over the course of an evening (in general) rather than a particular one.

Why is the word order … drikker jeg … and not … jeg drikker …?

This is the Norwegian V2 rule (verb-second). When you start the sentence with something other than the subject (here: I løpet av kvelden), the finite verb (drikker) must come second, and the subject (jeg) moves after it:

I løpet av kvelden (1st position) + drikker (2nd) + jeg …

If the subject comes first, you get normal order:

Jeg drikker bare vann i løpet av kvelden.

Can I move i løpet av kvelden to the end of the sentence?

Yes:

Jeg drikker bare vann i løpet av kvelden.

Both are correct; the difference is mainly focus/flow. Putting I løpet av kvelden first often sets the time frame as the topic before saying what happens within it.

What tense is drikker, and can it refer to the future?

Drikker is present tense. In Norwegian, the present tense is used not only for habits but also very often for planned/expected near-future actions, depending on context:

Habit: I løpet av kvelden drikker jeg bare vann (as a general rule for evenings like this)
Plan/decision: same sentence can also mean you’ve decided that this evening you’ll only drink water

Context usually makes it clear.

What exactly does bare modify here, and where else could it go?

Here, bare means only/just, and it most naturally modifies vann: you drink only water.

Placement changes emphasis:

… drikker jeg bare vann = only water (not other drinks)
… bare drikker jeg vann = only then / just (less natural here; sounds contrastive)
Jeg bare drikker vann can sound like I’m just drinking water (nothing else is going on), depending on context

So the given placement is the standard one for only + object.

Why is it vann with no article?

Vann is usually a mass noun (uncountable) meaning water in general, so it often appears with no article:

Jeg drikker vann.

If you mean a (serving of) water (like a glass/bottle), Norwegian can use et vann:

Jeg tar et vann. = I’ll have a water.

With bare vann, the article-less mass-noun reading is the normal one.

Is i løpet av interchangeable with på kvelden or om kvelden?

Not exactly:

i løpet av kvelden = during the evening / over the course of the evening (focus on what happens within the time span)
på kvelden = in the evening (more like “at evening time”; often a general time-of-day frame)
om kvelden = in the evenings (often habitual/general: evenings as a recurring time)

So i løpet av is more about the duration and events unfolding within it.

Why is it av (of) in i løpet av—does it literally mean “in the run of”?

Historically, yes: løp relates to “run/course,” and i løpet av is like in the course of in English.
In modern Norwegian, you don’t usually analyze it word-by-word; it functions as an idiomatic time preposition meaning during/over the course of.

How would I negate the sentence (say I don’t drink only water / I don’t drink water)?

Two common options depending on meaning:

1) Not drinking water at all:

I løpet av kvelden drikker jeg ikke vann. = I don’t drink water during the evening.

2) Not only water (i.e., you drink other things too):

I løpet av kvelden drikker jeg ikke bare vann. = During the evening I don’t drink only water.