There are circumstances where having more than two sexes (sexes are distinct reproductive contributors, genders are cultural/behavioral/grammatical categories) could indeed have an evolutionary benefit. After all, the history of life on Earth proves that two sexes are better than one; combining genes from different individuals allows for offspring to differ from their forebears routinely rather than merely through infrequent mutations, and so it doesn't take nearly as long for new traits to emerge and species to evolve. Life on Earth was just single cells for three billion years, then a couple of those cells figured out how to make whoopee and evolution exploded, taking only half a billion years to get to the vast diversity of multicellular organisms we have today. It may be true that asexual organisms can reproduce faster than two-sexed organisms, but obviously that hasn't given them a decisive advantage. Rate of growth is only one factor in evolutionary success. The ability to adapt to changing environments is far more crucial in the long run, and two-sexed organisms are far more adaptable than one-sexed ones. So three- or four-sexed organisms might have an additional edge in adaptability, which might surmount the added logistical difficulties.
On the other hand, multiple sexes would be a safeguard against harmful mutation in, say, an environment with higher levels of radiation or mutagenic toxins than Earth has. With four copies of each gene, a harmful mutation on one copy would probably be "outvoted" most of the time and its potential harm would be nullified. You'd need two or three mates with the same mutation for it to be expressed, and the odds of that would be low.
In real life, fungi, slime molds, and some bacteria are considered to have multiple sexes, but the definition of "sexes" is a bit loose there and not exactly analogous to vertebrate sexuality. Here's a thread I found on the subject:
http://boards.straightdope.com/sdmb/showthread.php?t=180007
The thread points out that the logistical difficulty in gathering enough mates together can be surmounted if the species can store a partner's gametes for later use, so that all the genetic contributors don't have to be in the same place at the same time.
^I agree. TNG's "The Chase" established that the ancient humanoids "seeded" humanoid life throughout the galaxy. Maybe they got bored making two-gendered species and decided to do something different with the Andorians.
Unlikely. The First Humanoids seeded the primordial soup of ancient worlds four billion years ago. If Andorian evolutionary history is anything like humans', they would've emerged less than one million years ago. There would've been a huge amount of evolutionary history between when the FH seeded Andoria and when the Andorian species emerged, so it's unlikely that the FH could've been responsible for any of the "detail work" in their evolution. Maybe some more recent superrace could've intervened in their evolution, but the FH were gone far too long ago.