Do fairies exist in the tolkien universe as apparently one of the tooks married one and it was from that point that they had adventures?

I think the fairies and other mythical creatures appeared in Tolkien's world as such, but they became Maiar when Tolkien refined his stories. Note that many Maiar are often either nature spirits or elemental spirits. Most of the fire spirits became Balrogs.

I honestly think that some Hobbits are just adventurous, and the fairy theory is something the Sackville Bagginses made up.

In this case, 'fairy wife' can be read as 'Elven wife', but the rumor is also characterized as absurd and should not be taken seriously. However, this might hint that the Tooks are Falohide hobbits, taller and slimmer than other hobbits, with fairer skin and hair.

That said, we don't know much about such beings as Tom Bombadil, the River-woman and Goldberry the River-woman's daughter. Goldberry and her mother, at least, might be akin to what we think of as fairies.