It is possible that someone who has high IQ also has high EQ. Just remember that this is not something that is automatic. It does not mean that just because you have a high IQ, your EQ is going to be high too. In fact, there are a lot of people with high IQ who have inversely low EQ.
This might be because they are able to think in a very rational manner. They do not use their emotions when they are making decisions. Some people may think that they are cold and that they do not care. Their IQ may have caused them to be wired that way.
Not necessarily. These 'quotients' are independent of each other. EQ is closely linked with personality and IQ is not. That said, there are obviously many people high in both quotients and sadly many low in both, too.
Nurture, upbringing, may improve functioning in both respects but we are born with a certain potential and we tend to live that out. However, it is possible to inhibit the growth of true potential and poverty and poor opportunity in life does this.