Learn Hardcore German: Ohne Wasser kann ich die Tablette kaum schlucken. - Without water I can hardly swallow the pill.

Ohne usually takes a noun in a general sense without an article when you mean “without (any) …” in general. So Ohne Wasser means “without water (in general / without any water).”
You can use an article if you mean specific water:

ohne das Wasser = without the water (a particular water you’re talking about)
ohne Wasser = without water as a substance / without any water

What case does ohne take, and how can I tell?

Ohne is always followed by the accusative case.
Here, Wasser happens to look the same in nominative and accusative (many neuter nouns do), so you can’t “see” it on the noun form itself. But the rule still applies: ohne + Akkusativ.

Why is the verb at the end in Ohne Wasser kann ich … schlucken?

Because the sentence starts with something other than the subject (Ohne Wasser is placed first for emphasis/topic), German uses verb-second (V2) word order in main clauses:

Position 1: Ohne Wasser
Position 2 (finite verb): kann
Then: ich …
The other verb form (infinitive) goes to the end: schlucken

So it’s not “verb at the end” overall—only the infinitive goes to the end because of können.

Why is it kann ich and not ich kann?

Both are possible, depending on what you put first:

Ich kann die Tablette ohne Wasser kaum schlucken. (subject first)
Ohne Wasser kann ich die Tablette kaum schlucken. (fronting Ohne Wasser for emphasis)

When you move something to the front (position 1), the finite verb still must stay in position 2, so ich comes after kann.

What’s the difference between kaum and nicht here?

kaum means “hardly / barely”—so you can swallow it, but it’s very difficult.
nicht would mean “not” (or “cannot” in a broader sense), which is stronger:

… kann ich … kaum schlucken = I can barely swallow it.
… kann ich … nicht schlucken = I can’t swallow it (at all).

Where does kaum go in the sentence, and could it go elsewhere?

kaum is an adverb and commonly sits in the “middle field,” often just before the infinitive phrase or near what it modifies.
Ohne Wasser kann ich die Tablette kaum schlucken is very natural.

Other placements are possible but shift emphasis:

Ohne Wasser kann ich kaum die Tablette schlucken. (more like: it’s the tablet specifically that’s hard to swallow—less natural here)
Kaum kann ich die Tablette ohne Wasser schlucken. (fronting kaum is possible, more literary/emphatic)

Why is it die Tablette (accusative), and how do I know it’s not dative?

Because schlucken (“to swallow”) takes a direct object, which is accusative.
So die Tablette is the thing being swallowed → accusative.
Dative would be used for certain verbs or indirect objects (e.g., “give someone something”), but not for schlucken.

Is Tablette always feminine? How do I know the gender?

Yes, Tablette is feminine, so it’s die Tablette (singular), die Tabletten (plural).
There’s no reliable “English-like” guess here; it’s a loanword but still has fixed gender in German. In practice, you learn it with the article: die Tablette.

Could I say Ohne Wasser kann ich kaum die Tablette schlucken or … die Tablette schlucken kann?

Ohne Wasser kann ich kaum die Tablette schlucken is grammatically fine but can sound like the emphasis is on die Tablette (as opposed to something else). The original order is more neutral.
Ohne Wasser kann ich die Tablette schlucken kann is wrong (double finite verb).
If you mean a subordinate clause (with dass, etc.), then you’d put the finite verb at the end:
…, weil ich die Tablette ohne Wasser kaum schlucken kann.

Why is Wasser not plural? In English we might say “without water” too, but is there a plural?

Wasser is normally an uncountable noun in German too, so singular is standard: Wasser.
Plural Wässer exists but is used for “different kinds of waters” (mineral waters, bodies of water, etc.), not for drinking water in this context.

Would German speakers ever use ohne Wasser at the end instead, like Ich kann die Tablette kaum schlucken ohne Wasser?

That word order is generally not idiomatic in standard German. You’d usually place ohne Wasser earlier:

Ich kann die Tablette ohne Wasser kaum schlucken. (very normal)
Ohne Wasser kann ich die Tablette kaum schlucken. (also normal, more emphasis on “without water”)

Putting ohne Wasser at the very end is possible in some contexts for afterthought/emphasis, but it often sounds marked or spoken/fragment-like.

What’s the difference between schlucken and runterschlucken here?

schlucken = to swallow (neutral).
runterschlucken = to swallow down, often more colloquial and a bit more vivid/emphatic.
In a normal statement like this, schlucken is the straightforward choice, especially in more neutral/formal contexts.