The Clear Fund is Givewell's registered name as a charitable foundation, for legal reasons. Jumping through bureaucratic hoops is sufficiently difficult that doing so over, or even changing the name of a foundation, is difficult. However, there apparently aren't restrictions on having 'Givewell' as the public-facing title of the organization, or a subsidiary, or something like that. I imagine the name comes from when Holden Karnofsky and Elie Hassenfeld started it as a project to discover charities with transparent operations, hence "Clear" Fund, before they re-branded and took on the project of fully fledged charity evaluation.

I have no idea, but it's the charity you get when you give in Givewell's EIN.

The Givewell website also has this as the footer:

GiveWell, aka The Clear Fund (a tax-exempt 501(c)(3) public charity) 182 Howard Street #208, San Francisco, CA 94105

[Bolding mine]