Whether humans "dominate" the Federation is unclear. The head of the Federation has been shown as a non-human twice in Star Trek history (Jaresh-Inyo, a Grazerite, and Ra-ghoratreii, an Efrosian) and human twice (Jonathan Archer and Hiram Roth). This is too small a sample size to draw any solid conclusions about the ratio of human to non-human presidents, but it does suggest the possibility that the head of state isn't always human (as would be expected given the democratic nature of the Federation).

It's also been suggested in at least ENT, TNG, and DS9 that there are many non-human and non-Earth-based starships in starfleet. While integration might be preferred given the goals and ideals of Starfleet and the Federation, there are practical advantages of having a crew that is mainly comprised of species from a single world:

 - The physiology of different species can vary greatly, and their environmental as well as ergonomic needs are different. It would be easier to crew starships with lifeforms that have compatible environmental needs and who can be accommodated by a shared ergonomic design of equipment.
 - Species that already live on the same worlds are accustomed to living and interacting with one another. Cultural differences are therefore less likely to come as a surprise or interfere with teamwork. Sharing a common thread can also bond the crew and strengthen comradery.
 - It's just more convenient to have ships based in Earth to be crewed primarily by inhabitants of Earth. The same would be true of starships based in other planets.

As for why Earth is the leading member (and in effect the capitol) of the Federation, that comes down to Earth being one of the founding worlds and the one that, as Mark Rogers points out, was the peacemaker that really brought all of the founding powers together to begin with.

So the bigger question is, *why isn't Earth a more integrated society?* Given the Federation's aims and ideals, you would expect to see lots of cross-species immigration between Federation member worlds. However, the only planets that seem to have integrated populations are Risa, a resort planet of the Federation, and Farius Prime, a neutral world that is in the lawless borderlands.