Well, as for the diplo penalty, if you don't plan on being friends with that AI, and plan on eventually taking them out anyway, doesn't really matter.
As for razing and resettling, you have to judge whether it is more worthwhile to resettle, or to keep. If the city is very small, you would tend more to raze. Is the city going to be a big help in the short run in it's current place? Are the long term advantages of a more ideal spot worth the work of building a settler and extra time to grow? Each case is a little different, and you just have to keep these things in mind. Another question to ask is whether the current city is going to interfere with a later or currently settled or captured city, resulting in one city or the other having stunted growth.
A lot of the time, in the early game, where a settler is a big investment, I'll tolerate a slightly non-ideal placement just because it's a city I don't have to build. This commonly happens with barb cities. Of course if the city location really is crap, raze away.
Keep in mind that food is king for city growth, and if by moving a city 1 square, you can get say 6 surplus food compared with 3, It won't take long for the new city to surpass what the old would be.