Well, a couple of answers that I'll reserve the right to revise as time goes on.
No. Negative mass is not possible. Certainly no such thing has ever been observed. Dirac predicted the existence of anti-matter as an interpretation of the negative energy eigenvalue solutions to the Schroedinger equation. I beleive ( and I need to verify this) that there are no such equivalent solutions to Einstein's GR equations or to Hamiltonian formulations of classical mechanics.
Although photons are massless, they do feel the influence of gravity. In GR, gravity couples not to an object with a non-zero rest mass, but to an object with momentum and energy. Photons may have a zero rest mass, but they do have momentum and energy, and feel the influence of gravity.
This isn't to say the idea of negative mass is totally off the wall (only partially off the wall). It might explain some aspects of dark energy (although this is outside my area of expertise). The main argument against it is that:
A) It would require a total reformulation of GR that is not obviously necessary.
B) There is no observational evidence that would suggest it.
"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind." - William Thompson, 1st Baron Lord Kelvin
"If it was so, it might be, and if it were so, it would be, but as it isn't, it ain't. That's logic!" - Tweedledee
This isn't right. This isn't even wrong. - Wolfgang Pauli