In two of the questions reference is made to Richard Rorty. I think that this is an unfortunate example for the questions' purposes. There are several reasons for this.
Rorty does refuse to develop a teleologically oriented notion of philosophy as coming to a final truth, but foundations can be above as well as at the end or the beginning. Indeed I'd argue that such meta-theories (though they may be about language or signs rather than about being explicitly) are more or less unavoidable. (For the less, see the chapter on irony in my Postmodern Sophistications .
- Rorty's definition of philosophy does not pretend to be a characterization of what philosophy has traditionally thought of itself as doing. He thinks that the conversation definition is a proposal for a new self-understanding of philosophy. (It may also be a characterization of what was really going on despite the traditional self-understanding, but that is a different issue. It's not what Rorty is worried about, though it's closer to what I had in mind.)
- Rorty's recent work, taken in its full sweep, is much more traditional philosophy than it might appear to be. That is, his views about philosophy as ongoing conversation are themselves based on a quite formal and rather rigorously argued foundation (a meta-theory about language and its structure and function) about language and truth. He adapts his rhetoric to his audiences, but his work since Philosophy and the Mirror of Nature does depend on ideas and arguments developed out of Quine and Sellars and Davidson. That meta-theory is argued in standard linear philosophical fashion. Some critics think that Rorty is committed a version of transcendental philosophy via a kind of very formal semantics. Rorty disagrees about this, though to me his conclusions do depend on such a foundation.