From the studies I've read, IQ and EQ scores tend to have a high positive correlation with each other, which suggests they're unified by a mechanism somewhere along the line. A study I read last week, actually, found that gifted folks typically exhibit comparable emotional nuances. This is not surprising to me since IQ scores attempt to determine one's intelligence by measuring their ability to manipulate abstraction, also known as g. The distinctions between IQ, intelligence, and g are important because they are separate constructs: the first is a measurement, the second is the purpose of the measurement (social connotations aside), and the third is the mechanism on which the measurement attempts to operate. The first two are, I believe, skewed images of the third. Using an ocular analogy, g is the lens and IQ/EQ are different wavelengths of light filtered through it to form an image of the reality swirling around us. One data stream is impersonal and universal (IQ), another is personal/meaningful and dynamic (EQ). They are different manifestations of the same thing; at the basest level, patterns are picked up and stored by the brain, where they are combined and recombined to construct perceptual webs in a manner similar to how genes are translated and altered to produce useful biology.
My neuroscience professor provided an interesting bit of data near the beginning of the school year: IQ is related to the density of certain cortical gyri (I don't remember which). In a literal sense, size matters ^.^ Since brain development is mostly genetic through the production of proteins and neurons, it's quite possible that IQ follows suit, which twin studies have been suggesting for decades. On the flip side, EQ scores may also be partly inborn. An essential part of the score is emotional empathy, the ability to connect with and understand others. My knowledge of it is incomplete, but the MAOA gene produces an enzyme attributed to deliberation, displays of empathy, and self-control. Popularized as the "warrior gene", monoamine oxidase A (MAO-A) is an enzyme produced by the MAOA gene that influences one's willingness to interact with and care for other people. Deficiencies of this enzyme incite aggression and impulsive behavior in mice and humans when under stress, and are found in the vast majority of American criminals, particularly sociopaths, who are trademarked by a lack of empathy. There is also the matter of the limbic system, which processes emotion. I cannot recall all of the neural pathways, but the cingulate cortex in particular is responsible for creating and processesing emotions, as well as learning and memory (in tandem with the hippocampus); collectively, it's responsible for associating actions, such as hugging, to positive emotional rewards, like the release of oxytocin and dopamine to produce and reinforce feelings of happiness. Another example would be the act of smiling, which is a prosocial cue learned by the firing of mirror neurons in the brain that tell us to imitate the behavioral information it receives from other people. Since neurons are cells, they are created by proteins scripted by DNA. This further suggests that the ability to relate to others ("if they're doing it, I should do it too because I'm one of them") and display empathy is partially genetic and partially acquired. We have the hardware, it just needs to be used.
Now that I think about it, and provided that it could be verified, I'd love to do a longitudinal study of all the personality types' brain activity as they go about their lives and mature. It would be interesting to see if any patterns of dendritic growth to particular spheres occur as a function of personality.
Humility disclaimer: I'm still a student and have not parsed through all the information on either IQ or EQ. If I'm wrong about something, please state why and provide relevant correction materials, if available. Learning is fun.
Now to answer the OP's actual questions.
1) Do you know and/or care about your EQ?
No, and I would not care for the number if I did. I'd want to understand how the mechanism works and how it affects my behavior in order to improve myself. Numbers are arbitrary; motion is not.
2) If you have a high EQ, do you take pride in that?
Not applicable, but, again, the numbers don't mean anything to me. They are statistics and prone to change across space, time, and thought/methodology.
3) Do you think IQ or EQ is more important?
I think they're equally important for healthy functionality and should be integrated. While I believe they are fundamentally similar, their expression differs. They use different kinds of data to perform different tasks, therefore, one cannot be evaluted in terms of the other and each is needed to fulfill particular needs.