At least in historically Christian communities, it has negative connotations theologically. In the Christian understanding of the creation narrative Eve eats the forbidden fruit, bringing sin into the world and causing all sorts of bad things (i.e. original sin).
In Jewish communities those connotations don't exist, since Adam and Eve are seen both negatively to an extent, and more importantly their disobedience is not seen to have the same theological repercussions as in Christianity. As a consequence Eve in Hebrew, which is Chava/Hava/Chava/Chave is a fairly common woman's name and has been for quite a while.
I am not sure it is.
At least, I think it must depend on your location. I live in Iceland, and over here the name "Eva" is about 6 times more common than "Adam", and that is just as a fist name. If you include it as a second name, it is even more popular.
(Source: the Icelancic national registry, 2015)
Eve isn't popular but Eva, the Latin equivalent, has been quite popular in the United States in the past, often being among the hundred most popular names for girls (it's also enjoying a resurgence in popularity, perhaps due to the increase in the size of the Latino population). In fact, if you compare it to Adam, it's actually been more popular overall.
One issue as well is that where "Adam" is somewhat consistent linguistically, appearing more or less the same in Hebrew, Arabic and English at least, "Eve" is not, appearing as "Haw'a" in both Hebrew and Arabic, and as others have pointed out, as "Eva" in various other languages.