Learn Hardcore Japanese: nodo ga itai toki ha mizu wo takusan nomimasu. - When my throat hurts, I drink a lot of water.

喉は痛い sounds like “As for my throat, it hurts (but maybe other parts don’t)” – there’s a contrastive or topic feel.
喉が痛い simply states that the throat is what hurts.

In this sentence, the neutral and most natural choice is 喉が痛い.

Why is it 痛い and not 痛いです before とき?

Before とき, you normally use the plain form of the verb or adjective, even in polite sentences. Politeness is shown in the main clause, not in the とき clause.

So you say:

喉が痛いときは、水をたくさん飲みます。
- Plain adjective 痛い
  - とき
- Polite verb in the main clause (飲みます)

Using 痛いですとき is unnatural.

Pattern to remember:

Verb/adjective in plain form
- とき
  - (main clause, polite or casual)
    - 暇なとき、本を読みます。
    - 忙しいときは、連絡できません。
    - 時間があるときに、電話します。

What exactly does とき do here, and why is there also a は after it (ときは)?

とき (時) is a noun meaning time. The phrase before it modifies it, so:

喉が痛いとき literally = the time when my throat hurts → when my throat hurts.

Then は marks that whole phrase as the topic:

喉が痛いときは
= As for the times when my throat hurts,
= When my throat hurts,

This X ときは、Y pattern is very common for talking about what you do in a certain situation:

暇なときは、映画を見ます。
困ったときは、友だちに相談します。

You could also say 喉が痛いとき、水をたくさん飲みます。 (without は) and it is still correct.
With は, the situation 喉が痛いとき is highlighted as the topic or condition you are talking about.

Why is 飲みます (non‑past) used, and not a past tense like 飲みました?

The non‑past form in Japanese (for verbs: 飲む / 飲みます) covers:

present actions
future actions
habitual actions (what you usually/always do)

Here, the sentence describes a habit:

喉が痛いときは、水をたくさん飲みます。
= When my throat hurts, I (usually) drink a lot of water.

Using 飲みました would make it sound like you are talking about specific past times when your throat hurt:

喉が痛いときは、水をたくさん飲みました。
= When my throat hurt (on those occasions), I drank a lot of water.

So non‑past 飲みます is correct and natural for a general habit.

Why is it 水をたくさん飲みます and not たくさん水を飲みます? Are both correct?

たくさん is an adverb meaning a lot / many / much. With verbs like 飲む, both of these are possible:

水をたくさん飲みます。
たくさん水を飲みます。

Both are grammatically correct and commonly used. The nuance is slightly different:

水をたくさん飲みます。
- Emphasis a bit more on how much you drink.
たくさん水を飲みます。
- Emphasis a bit more on water as the thing you drink a lot of.

In everyday conversation, they are very close in meaning. Your original 水をたくさん飲みます is perfectly natural.

What does the particle を do after 水?

を marks the direct object of the verb.

水 = water
水を飲みます = I drink water.

So in 水をたくさん飲みます, 水 (water) is what is being drunk, and を shows that it is the object of 飲みます.

In casual speech, を is sometimes dropped in fast conversation:

水たくさん飲む。

But in correct standard Japanese, especially in writing or polite speech, you keep を.

Why don’t we say 私の喉が痛い? How do we know it’s my throat?

Japanese often omits possessives like 私の when it is obvious from context, especially for:

one’s own body parts (喉, 頭, 手, 足, etc.)
one’s family members
things clearly belonging to the speaker

If you say 喉が痛い, it is naturally understood as my throat hurts unless context says otherwise.

You would only say 私の喉 if you really need to contrast or emphasize ownership, for example:

私の喉が痛いんじゃなくて、弟の喉が痛いです。
= It’s not my throat that hurts; it’s my younger brother’s.

In normal statements about yourself, 私の is usually omitted.

Could I just say 痛いときは、水をたくさん飲みます。 without 喉?

Grammatically, yes:

痛いときは、水をたくさん飲みます。
= When it hurts, I drink a lot of water.

But then what hurts is unclear. It could be your head, stomach, throat, etc.

If the conversation has already made it clear that you are talking about your throat, then omitting 喉 is fine and natural:

A: 最近、どう？
B: 風邪をひくことが多いんだ。
A: そうなんだ。
B: 痛いときは、水をたくさん飲みます。

Here, from context it might be clear, but in isolation, 喉が痛いときは is more precise.

Can I say 喉が痛いです instead of 喉が痛い? How is 喉が痛む different?

喉が痛いです
- Polite version of 喉が痛い.
- As a standalone sentence, it’s perfectly natural:
  - 喉が痛いです。= My throat hurts.
Before とき, you normally keep it plain:
- 喉が痛いときは、水をたくさん飲みます。 ✅
- 喉が痛いですときは、水をたくさん飲みます。 ❌ (unnatural)
喉が痛む
- 痛む is a verb meaning “to ache, to hurt”.
- 喉が痛むときは、水をたくさん飲みます。 is grammatically correct, a bit more formal/literary.
- 喉が痛い (い‑adjective) is more common and conversational.

So:

Use 喉が痛い (plain) in the とき clause.
Use 痛いです in standalone polite sentences, not directly before とき.

Does 喉が痛いときは mean “whenever my throat hurts” or “if my throat hurts” or “when my throat hurts”?

It can cover all of these ideas, depending on context. X ときは、Y often has a general, habitual meaning:

喉が痛いときは、水をたくさん飲みます。
= When/whenever my throat hurts, I drink a lot of water.

It can also feel close to a conditional if in English:

If my throat hurts, I drink a lot of water.

Japanese does not sharply separate when vs if here; とき plus non‑past generally describes situations in which X is true, and what you do in those situations.

How would I change this sentence if I want to say “When my throat hurts, I should drink a lot of water”?

You can keep the ときは part and change the second half to express advice / obligation:

喉が痛いときは、水をたくさん飲んだほうがいいです。
= When my throat hurts, I should / it’s better to drink a lot of water.

Other options:

喉が痛いときは、水をたくさん飲むようにしています。
= When my throat hurts, I make a habit of drinking a lot of water.
喉が痛いときは、水をたくさん飲まなければなりません。
= When my throat hurts, I must drink a lot of water.

The ときは part stays the same; you just choose a different expression in the main clause.