Learn Hardcore Chinese: nǐ yào yì bēi shuǐ ma?

Before a 1st, 2nd, or 3rd tone, it becomes yì: 一杯 (yì bēi), 一年 (yì nián), 一碗 (yì wǎn).
Before a 4th tone, it becomes yí: 一个 (yí ge), 一份 (yí fèn).
When counting numbers in isolation, it’s yī.

Why do we need 杯? Why not just say 一水?

Chinese needs a measure word (classifier) between numbers and nouns. The pattern is Number + Measure + Noun. For liquids you usually use a container as the measure:

一杯水 (yì bēi shuǐ) a cup/glass of water
Compare: 两瓶水 (liǎng píng shuǐ) two bottles of water, 一碗汤 (yì wǎn tāng) a bowl of soup

What’s the difference between 杯 and 杯子?

杯 is a measure word meaning “cupful/glass.”
杯子 is the actual physical cup. Examples:
给我一杯水。 Give me a cup of water. (measure)
我有一个杯子。 I have a cup. (object) You may hear 一杯子水, which colloquially emphasizes “a whole cup of water,” but the standard way is 一杯水.

Is 要 (yào) the best verb here? How is it different from 想 (xiǎng) or 想要 (xiǎngyào)?

要 = want/need; in offers/requests it’s common and neutral: 你要一杯水吗？
想 = would like/feel like; a bit softer: 你想喝点儿水吗？
想要 = “want” with a slightly stronger intention; also fine for objects.
In other contexts 要 can mean “be going to/need to” before a verb phrase: 我明天要去北京。

How do I make this more polite?

Use 您 instead of 你: 您要一杯水吗？
Add softeners like 点儿 (diǎnr) or use 喝: 您要不要喝点儿水？, 要不要来杯水？
In service contexts, 需要 (xūyào) can sound considerate: 您需要水吗？

Where does 吗 go? Do I need to change word order?

吗 sits at the very end of a yes–no question. Word order stays the same as a statement:

Statement: 你要一杯水。
Question: 你要一杯水吗？ Never put 吗 after a single word like 要; it modifies the whole clause.

Can I drop 吗 and just use rising intonation?

In casual speech, natives sometimes do that, but for clear, standard Mandarin it’s better to keep 吗 or use the A-not-A pattern. As a learner, prefer 吗.

What’s the A–not–A alternative to ask the same thing?

Use 要不要 or 想不想:

你要不要一杯水？
你想不想喝点儿水？ These are very common and sound natural.

How should I answer this question correctly?

Answer with the verb, not 是/不是:

Yes: 要。/ 要，谢谢。/ 好的，谢谢。
No: 不要。/ 不用，谢谢。 (literally “no need, thanks”)

Can I omit 一 and say 你要杯水吗?

Yes, in colloquial requests/offers it’s common to drop 一 before a measure word:

给我杯水。 (colloquial)
More careful/neutral speech keeps 一: 给我一杯水。

Should I use 吧 instead of 吗 here?

Different function:

吗 forms a neutral yes–no question: 你要一杯水吗？
吧 softens a suggestion/assumption: 喝点儿水吧？ (You should/How about having some water?)

Can I use 呢 here, like 你要一杯水呢?

Not by itself to form a yes–no question. 呢 often marks a follow-up or ongoing situation:

我不要。你呢？ I don’t want any. How about you?

How do I say two cups, or use other containers?

Two cups: 两杯水 (liǎng bēi shuǐ) — use 两, not 二, before measure words.
Other containers: 一瓶水 (yì píng shuǐ) a bottle of water, 一壶茶 (yì hú chá) a pot of tea, 一碗汤 (yì wǎn tāng) a bowl of soup, 一口水 (yì kǒu shuǐ) a mouthful of water.

Any tone-sandhi issues with 你 (nǐ) or 水 (shuǐ) here?

你 (nǐ) is 3rd tone, followed by 要 (yào) 4th tone, so it stays 3rd (no change). When two 3rd tones meet, the first becomes 2nd (e.g., 你好 is pronounced ní hǎo).
水 (shuǐ) is 3rd tone and is followed by neutral 吗, so you pronounce a “half-3rd” (low) on 水 in fluent speech.

What’s the difference between 吗 and 嘛?

吗 (ma) is the yes–no question particle: 你要一杯水吗？
嘛 (ma) is a modal particle used to soften, explain, or coax: 别急嘛。 They are not interchangeable.