In my experience I never hear the 'D'.
It's like it's dropped and you are left with "Pa Thai". Maybe it's because the letters sound the same and the flow of speak takes it away.
Either, really. And you’ll hear Thai people pronounce it either way (depending on their English reference - usually British), since locals will not call it “Pad Thai”.
Not sure what you mean by this. This is the name in Thai and "ผัดไทย" ([pʰàt tʰāj] in IPA) is what is written on shops and carts where it's sold.
The vowel in "pad" is in between the two options given in the question above. Unlike English, where accents primarily change the vowels, vowels stay quite constant across accents in Thai.
The /a/ in Thai is a central vowel. This is different than English /a/, which is a front vowel. And /o/ is the open back vowel. That is why it falls between English ⟨pad⟩ and ⟨pod⟩. And if you don't speak a language with such a vowel, any particular utterance will probably be mentally mapped to one or the other.
Pad as in iPad?
Or closer to pod?