"Prince Andrew, looking again at that genealogical tree, shook his head, laughing as a man laughs who looks at a portrait so characteristic of the original as to be amusing."
This sentence is large and awkward, but it seems to be grammatically sound. As you suggested, it looks like it is taken from a piece of literature, probably fiction. The simplest version of the sentence that only contains the subject and the subject's action looks like this:
Prince Andrew shook his head.
On to the OP's questions:
1. Does " looking again at that genealogical tree" equal "who was looking again at that genealogical tree" ?
Not exactly. Your version can be made into a complete sentence ("who" is the subject). The version the author used is a parenthetical phrase being used as an adjective -- namely, a participle phrase. "Looking again at that genealogical tree" describes Prince Andrew as a complete phrase. The two phrases' meaning is the same, but grammatically they are different.
2. Does "laughing as a man laughs..."equal " while he was laughing .."
No. "Like" and "as" in literature are important to look for, because they usually are markers that denote the start of a comparison. These specific comparisons are called "similies" by authors. This particular example is more complicated because the author is using "as" for two different purposes in this sentence.
laughing as a man laughs...
Here, "as" is used to compare "Prince Andrew's laugh" to "the same kind of laugh that a man [meaning any man] would have while looking at a portrait which is so characteristic [meaning "similar"] of the original as to be amusing."
...of the original as to be amusing.
Here, "as" is used in a way similar to "that" -- it's so characteristic of the original that it is amusing. It is an uncommon and somewhat dated usage.
3. why is there a noun "laughs" after the man? and the "who"
"Laughs" isn't a noun. "Who" refers to "a man." Could you clarify why you think that's wrong?
4. is " characteristic" here anadjective or noun? and i found a word "so" placed before it.so i guess the "characteristic" is an adjective. but there is no any prepositions before it.. or maybe it is the structure of "so + adj + as to be"..???
It's an adjective, meaning "similar" or "resembling."