Learn Russian: Дайте мне чек, пожалуйста.

Дайте is the polite or plural command form of the verb meaning "to give". The informal command is дай (used with friends). We add -те to make it formal, which is perfect for speaking to a cashier or waiter.

Why do we use "мне" instead of "меня" here?

Мне is the Dative case of я (I). In Russian, the recipient of an action (the person you give something TO) takes the Dative case. Меня is the Accusative case, which would mean you are the object being given!

Since "чек" is the direct object being given, shouldn't its ending change?

It is indeed the direct object and is in the Accusative case! However, because чек (receipt) is an inanimate masculine noun, its Accusative form looks exactly the same as its dictionary form.

Why is the letter "й" in "пожалуйста" silent when people say this sentence?

Even though it is spelled пожалуйста, the й is dropped in natural speech. You will usually hear native speakers pronounce the word as "пажалста" (pa-zhal-sta).