in the question "What's your last name for me?" asked by a clerk for a registration, does the prepositional for me soften it?
It's ungrammatical and doesn't really make sense. "for me" would normally follow you asking someone to do something, like "Can you spell your last name for me?".
If you want to "soften" (which which I take it you mean 'make more friendly or welcoming') "What's your last name?" then you can say "What's your last name, please?" or "Please can you tell me your last name?"
PS that video is creepy.