What does the 푸 in 푸대접 mean?

ANSWER: I'm not sure, but I have a theory. 

But first, I have a few questions about the Korean language: 

1. What is the difference between 대접(待接) and 접대(接待), besides just the switching of the Chinese characters? My dictionary says that both 대접하다 and 접대하다 can be translated as “to treat” or “to entertain,” so when do Koreans use one and not the other?
2. The Chinese character 不 (부) means “not,” so if 대접 (待接) means “hospitality,” then 부대접 should mean “inhospitality,” but Koreans do not say 부대접; instead they say 푸대접. Why? Does 푸 mean “not”?
3. If Koreans can use 푸대접 to mean “cold or unkind treatment” or “inhospitality,” why can’t they also say 푸접대?
4. The Korean word 푸접없다 means “unfriendly or cold and distant treatment.” That implies there should be a 푸접있다 or at least a 푸접, and, indeed, there is such a word. 푸접 means “to treat someone in a warm and friendly way,” and that implies that the 푸 in 푸접 means “warm and friendly. So my final question is the following:
   
   Why does 푸대접 mean “cold, unfriendly treatment” instead of “warm and friendly treatment," considering that the 푸 seems to mean "warm and friendly"? Could 푸 come from 품, which means "to hug someone"?
   

I think 푸접 is just an abbreviated form of 푸대접 and that, somewhere along the way, Koreans got the word 푸대접, meaning "warm and friendly treatment," confused with 부(不)대접, meaning "inhospitality." Like Americans, Koreans sometimes confuse the sounds of 푸 and 부.

Moreover, I think 푸대접 was once 품대접, meaning "hospitality that begins or ends with hugs." After all, hugs are warm and friendly.