Ok, there are a few ways of looking at it.
Iluvatar is the one who created the Ainur, the Maia, and the Elves. Supposedly all eleves are related because Iluvatar created all of them.

Now some people believe that the elves have sexaul intercoarse and create their families that way.

It really depends on how you look at it. I will do some reasearch and get back to you on this subject.

To anyone still wondering about this question, the answer is, "no". It's not written anywhere by Tolkien.

He did, however, write much about Arwen's genealogy and her family's history. Tolkien does not give much information about Legolas or his family at all, as he is a relatively minor character in his overall work.

As such, it's impossible to tell for certain, since no name is given for Legolas's mother or grandmother. Could either of these nameless elves be a missing link between the families of Legolas and Arwen? While it is possible, it's not likely.

Even if they were related, they would definately not be first cousins, since these relatives are accounted for.

three main houses of elves are mentioned, the Noldor, the Vanyar, and the Teleri. arwen is of the of the noldor along with other notables: elrond, gil-galad, etc. legolas is of the sindarin, who are the descendants of the teleri. since the three kindreds of elves were formed from the eldar, one of two groups of elves before that, they are in fact related, but so distantly, that it almost makes them completely alienated.

not in any way Tolkien mentioned....I didn't find any such thing in the book. even when the fellowship was in Rivendell, nothing mentioned happened b/w them....Legolas was an elf of honor greater than Arwen, if I was to be asked about them

yes

It is true that in all three Lord of the Rings books Tolkien never mentions that Arwen is related to Legolas and we never really get an entire background on each of them. We are left to assume that through some distant way they are related though because of how the Elves work. They're divided up into four major groups, however, that does not mean that these groups are not related to each other. More than likely Arwen and Legolas are related in some form or another.

Not that I know...I just read the series and it did not say anything about that. However, all elves like man in real life, are all related to each other but probably so distantly related, so I would say no...