Learn Hardcore Russian: Пожалуйста, дайте мне ещё один карандаш.

Пожалуйста literally means “please” when making a request. In Russian it’s a particle of politeness and can appear at the beginning, middle, or end of a request with no change in meaning.
Examples:
• Пожалуйста, дайте мне карандаш.
• Дайте мне карандаш, пожалуйста.

Why is дайте used here instead of дай?

Дайте is the formal or plural imperative of дать (“to give”).
Use дайте when speaking politely to strangers, elders, or groups.
Дай is the informal singular imperative, used with friends, family, or children.

What case is мне, and why do we use it?

Мне is the dative case of я (“I/me”). In Russian, the person who receives something (the indirect object) takes the dative case. Here you’re asking someone to give something to you, so you say мне.

Why do we say ещё один карандаш rather than just ещё карандаш?

Ещё means “more” or “again,” but without a numeral it doesn’t specify quantity. Ещё один means “one more” or “another one.”
• еще карандаш = “more pencil” (ungrammatical)
• ещё один карандаш = “one more pencil”

Can карандаш be in a different case, or is accusative the same as nominative here?

Карандаш is a masculine inanimate noun. In Russian, the accusative singular of masculine inanimate nouns is identical to the nominative. So as the direct object it remains карандаш.

Could I say Дайте мне ещё карандаш and leave out один?

No. Omitting один makes the phrase ungrammatical. You need один (or another numeral/determiner) after ещё to specify exactly one extra item.
If you wanted two more, you’d say Дайте мне ещё два карандаша.

How do you pronounce ещё correctly?

The letter ё is stressed and pronounced [yo]. The letter щ is pronounced [shch] (a soft, two-part sound). So ещё sounds like “ee-shchó” with the stress on the second syllable: [i-shCHÓ].