qǐng gěi wǒ yì zhī qiānbǐ, wǒ xiǎng zuò bǐjì.

Because 一 changes tone depending on the word that follows it.

Basic rule:

• 一 is normally yī when said alone or when counting.
• Before a 1st, 2nd, or 3rd tone, it changes to 4th tone: yì
• Before a 4th tone, it changes to 2nd tone: yí

Here, 支 zhī is a 1st-tone word, so 一支 is pronounced yì zhī.

So:

• 一支 → yì zhī
• dictionary form is still yī

支 is a measure word (also called a classifier).

In Mandarin, when you say one / two / three + noun, you usually need a measure word between the number and the noun.

Pattern:

• number + measure word + noun

So:

• 一支铅笔 = one pencil

Why 支? Because 支 is commonly used for long, stick-like objects such as:

• 笔 (pens)
• 铅笔 (pencils)
• some other slender objects

It is much more natural than using the general measure word 个 here.

In standard Mandarin, 一个铅笔 sounds unnatural and is generally considered incorrect.

With a number, Mandarin usually requires the appropriate measure word, and for 铅笔, that measure word is normally 支.

So the natural form is:

• 一支铅笔

A learner might hear 个 used very casually with some nouns, but for pencil, 支 is the normal choice you should learn and use.

The structure is:

• 请 = please
• 给 = give
• 我 = me
• 一支铅笔 = one pencil

So the pattern is:

请 + verb + person receiving + thing

More specifically: 请给我一支铅笔 = Please give me a pencil

Notice that Mandarin does not need a word like English to here.
English says give a pencil to me or give me a pencil.
Mandarin simply says 给我一支铅笔.

Yes, 请 here works like please and makes the request polite.

At the beginning of a sentence, 请 + verb is a common way to make a polite request:

• 请坐 = Please sit.
• 请看 = Please look.
• 请给我一支铅笔 = Please give me a pencil.

It is polite, but the overall tone still depends on context and voice.

One extra nuance: If you are asking someone to let you borrow a pencil, 给 may sound more like give/hand me. In real life, a more natural borrowing request is often:

• 可以借我一支铅笔吗？ = Can you lend me a pencil?

So your sentence is grammatical and understandable, but in some situations 借 may fit better than 给.

Both are possible.

• 请给我一支铅笔 = Please give me a pencil.
• 请你给我一支铅笔 = Please, you, give me a pencil.

In Mandarin, the person being addressed is often understood from context, so 你 is frequently omitted.

Adding 你 can:

• make the target clearer
• sound a little more direct
• sometimes add slight emphasis

So 请给我一支铅笔 is completely normal and natural.

想 can mean both think and want to, depending on context.

In this sentence, 想 + verb means:

• want to
• would like to
• intend to

So:

• 我想做笔记 = I want to take notes

If 想 means think, it usually introduces an idea or opinion, for example:

• 我想他不会来。 = I think he will not come.

So here, because 想 is followed by the action 做笔记, the meaning is want to.

Good question. 记笔记 is very common, and 做笔记 is also used.

Both can refer to taking notes, but there is a slight nuance:

• 记笔记 focuses more directly on writing down notes
• 做笔记 can feel a bit like making/producing notes

In many everyday situations, both are acceptable.

So:

• 我想做笔记
• 我想记笔记

Both are understandable and natural.
If your goal is to learn the most common textbook-style expression, 记笔记 is especially common.

Yes, 铅笔 is one word meaning pencil.

The characters are:

• 铅 = lead
• 笔 = pen / writing instrument

Historically, that makes sense because pencils were associated with lead, even though modern pencils actually use graphite.

So you should treat 铅笔 as a normal vocabulary word meaning pencil, not as two separate words in this sentence.

The comma separates two related clauses:

• 请给我一支铅笔
• 我想做笔记

The second part gives the reason for the request:

• Please give me a pencil, I want to take notes.

In speech, there would usually be a small pause there.
The comma helps show that these are two connected parts of one sentence.