So, Paul said we should love our enemies. Anybody else?
heaping "burning coals on his head" is often mistaken for allowing your enemy to feel guilty or something similar. Actually, the reference implies giving your enemy the means to cook food and keep warm.
I'm not inclined to agree with either interpretation. I think the verse presume you're enemy is in the wrong and that by returning malice with kindness you will either "convict" your enemy, cause a change of heart and make a friend; or return vengence to God's judgement where it belongs.
Regardless you are doing right by being kind.
No, Paul did not say this. My interpretation of the passage is that we should seek to do good to them.
I meet many people all day in my work. I try to be kind to each person, but I do not "love" them all except as terrestrial kindred. I reserve "love" for a very, very select few.