You probably guessed, but "cold diplomatic relations" makes sense to me.

To me, "I'm kidding with you" means that the speaker said something mean or untrue about the other person. Example: "You look ugly wearing that shirt." "Really?" "No, I'm kidding with you." "I'm kidding you" would be said if the speaker said something unbelievable that was not true. "I won a new car!" "Really?" "No I'm kidding you." However, in both cases you can also just say, "I'm kidding."

There may be some dialects that pronounce it the same, but where I live (Ohio in the U.S.) there is a difference. My lips spread out farther to the sides to produce the vowel in "men" than the vowel in "man."

Not always. "Whichever" implies that there is some sort of choice involved. "Eat whichever cookie you want." But you could never say "Eat whichever want" if you mean "you can eat anything in the house." If you just say "Eat whichever you want" the context has to be that there are some restrictions on what the person can eat and they can choose which one they want. If you say "eat whatever you want" it means the person can eat anything in the house, there are no restrictions. But you can also say, "Eat whatever cookie you want." So you can always use "whatever" but only "whichever" in a context where there is a limited amount of things for the person to choose.

Actually, I would probably say, "I'll pass on that" or something along those lines. I would not say, "I'll give up on that." As for what can go in the front: "On second thought" or maybe "Now that I think about it" are probably best. There isn't a idiom "on the second hand" but there is "on the other hand." However, this cannot be used in this context.

For me personally, I know that relations between China and the U.S. are going to be very important in the coming years. I want to be a part of that, either in the government or in the private business sector.

Hope means that you want something in the future. "I hope I pass my test." Want is just you would like something. "I want to eat a cookie." Wish is similar to hope, but usually it is not used for yourself. You could say, "I wish that you pass your test." but not "I wish that I pass my test."

I have never heard this before, what is the context?