Your name

Korean 'is' is not a separate word standing between two nouns, as in English. It is an ending glued to the noun it describes, and it comes last, so the sentence runs 'I, Sam-is'. After a consonant the 이 of 이에요 gives that consonant a vowel to lean on, which is why 샘이에요 is said sae-mi-e-yo. After a vowel the 이 has nothing to do and merges into the next sound, so 애나 plus 이에요 contracts to 애나예요. This split between nouns ending in a consonant and nouns ending in a vowel comes back again and again, starting tomorrow.

뭐예요 is 뭐 (mwo), 'what', plus the same 'is' as above, and 뭐 ends in a vowel, so 예요. That means the question and its answer share one frame: 이름이 뭐예요? is answered with 샘이에요, the unknown swapped for the name. Korean also avoids the word 'you' with people it does not know well; the words exist, but they sound too direct or even rude. So you ask about 'the name', and address people by name plus 씨, rather than saying you at all.

These phrases use the formal -ㅂ니다 ending of day 1's 감사합니다, because a first meeting is a slightly ceremonial moment. 잘 부탁드립니다 has no real English equivalent: it asks the other person to look kindly on you as the relationship starts, and it quietly assumes there will be one. Notice that 반갑습니다 and 반가워요 are one word with two endings, the same pair as 고맙습니다 and 고마워요 on day 1. When a word ends in ㅂ, as 반갑 and 고맙 do, the -요 form swaps that ㅂ for 워, which is why each pair looks less alike than it really is.