Can we add “just”, “almost” or “nearly” to the phrase “be about to”

You can, but not in that way.

If you wanted to say that you've come to know only a brief moment ago that the person you're talking to hates you, then the simplest way to change your original sentence to be correct is like this:

I've just realized you hate me.

However, you can use the phrase "just realize," as illustrated by the reply from the other person:

Did you just realize that I hate you?

Alternate version of the same question in past tense:

Have you just realized that I hate you?

I don't think you can use "just realize" outside of a question, though. Someone correct me if I am wrong on this.

It should be noted that despite the fact that the tenses are different, the meanings are virtually identical. The reason is the use of the word "just," which is used as an adverb to mean that it happened very recently. We understand that the realization happened only a second ago in both sentences even though the tenses are different.

“That is typical of you” is exactly what “You would!” means in this context. It is listed in some dictionaries, though it can be a little hard to find among the many other meanings of would; for example Cambridge lists it as

would modal verb (FREQUENCY) (…) disapproving used to ​suggest that what ​happens is ​expected because it is ​typical, ​especially of a person's ​behaviour:

"Madeleine called to say she's too ​busy to come." "She would - she always has an ​excuse."

This is closely related to the meaning explained just above: would can express something habitual. For example

He would always ​turn and ​wave at the end of the ​street.

means the same thing as “He always turned and waved …” but the use of would stresses the fact that it's a habit.

As for what you read in Urban Dictionary — don't mistake UD for a useful reference. A lot of what it contains is just what one person decided to add and doesn't mean that anybody else would understand. I, for one (admittedly not a native English speaker), have no idea what why “you would” is supposed to be annoying or this “I did forget”/“but really” business is supposed to mean (maybe it implies that “you would” was used sarcastically?).