"Give me, please, a napkin" is not correct English and should not be allowed. "Please give me a napkin" should be allowed, as this is correct English.
"Give me, please, a napkin," is correct English, though unusual. It seems to be no longer allowed, which is confusing.
Agreed. "Please give me a napkin should be allowed, at least as an alternate translation.
Why is "будь ласка" placed in the middle of the sentence? What is the rule there?
There is no special rule for that. "Будь ласка" can be placed at the end and beginning as well.
Without the specification in "Дайте" it can mean "give me" or "give us" (if someone is speaking on behalf of many people). It cannot mean "give him" because then it would be "дайте йому", we would not skip the pronoun in that case. We skip it only when we mean, give it to the one speaking, to me (or us).