Disclaimer: I only know two ENFPs, only one of which well. Also, I'm an INTx and exhibit only half of the general intP traits.
How does one know if he/she is considered to be a close friend of an INTP? How does one know if he/she is in the INTP's inner circle? What do INTPs do/say for/to their close friends that they wouldn't do/say for acquaintances?
Look at the things they tell you about love interests, people they like, worries they have. Then look at how many of their friends know. If that number is small, you're in the inner circle.
Generally, if the list is small, it's not because they just haven't told anyone yet: it's because we're not
going to tell many people.
What traits do INTPs look for close friends?
Perfection.

Sorry. This list is too long.
Which traits can INTPs NOT STAND in people?
When they breathe.

Sorry. This list is also too long.
What are the traits specifically in ENFPs that INTPs are annoyed by or dislike?
The first thing that comes to mind is indecisiveness. INTPs are known for being indecisive, but at least we can
come to some sort of conclusion. The ENFPs I've known have tended to be unable to form an opinion of their own, possibly for fear it'll hurt someone's feelings - though the simple act of being unable to do so is annoying.
For instance, if you take them on some new experience then ask them if they liked it, you'll typically be met with "I'm not sure. I mean, I might have, but that could also just be because I thought it was new and thought I might like it. But I'm not sure I thought I'd like it because I didn't know if it'd be fun or not, so that can't be it. Maybe I'm trying to tell myself that I liked it because I'd feel bad for not liking it. Yada yada yada." This is especially irritating when I actually do care whether or not they liked it (and I wasn't
just being socially polite).
Every ENFP I've seen has seemed to also feel they can just wish a better world into place and wish that everybody could just get along and things could be perfect, and seem to be unable to grasp reality.
Also, your tendency to believe anything, no matter how outlandish (such as "The Secret") is very, very irrational (and therefore annoying).
What are the traits specifically in ENFPs that INTPs admire/relate/ are attracted to?
I'm finding it hard to come up with specifics right now. The negative specifics are generally easier to remember (for me) than positive ones.
I know that we actually share an interest in typology: something which cannot be said of most people. We generally have a common interest in theories like this or other things involving understanding people - for different reasons of course, but it's still there.
I've found that ENFPs can be very, very awkward. They can often act like they like you, insist they don't, then spend weeks doubting whether or not they really do... Which makes things awkward.
And, when you
were close enough friends to actually be comfortable hugging them (a rare occurrence for me), this makes that too awkward. Then it gets really awkward when they want a hug and I can't give it to them because of the awkwardness. Then I feel bad because I'm making them feel bad.