The WMAP space probe allowed better measurements of the CMBR, from which the relative composition of matter at the time of the CMBR's emission could be determined (using methods I don't fully understand). Once it was known that the Universe was X% atomic nuclei, Y% photons, Z% dark matter, etc., this information implied certain bounds on how quickly the Universe was expanding during the inflationary epoch (10^-36 to 10^-32 sec after the Big Bang), since high rates of expansion can overpower the nuclear forces and prevent quarks from bonding to form protons/neutrons. The math for all of this is, unfortunately, far over my head. Sorry I can't be of more help.
I'm not fully understanding your next question. The origin of our light cone is here and now. Minkowski spacetime is an approximation that (almost) works in the absence of gravitating bodies. Once general relativity gets involved, certain features of Minkowski spaces start failing, like the fact that you can always reorient light cones so that they're parallel. This fails even in intergalactic space, where there is still a non-vanishing Weyl tensor effect due to Hubble Expansion. So, without getting too much into the detail, in the real world light cones are (forgive me) uncannily un-coney. Our past light "cone" collapses back in on itself in the distant past and converges on the Big Bang, but so do light cones everywhere in the Universe, even outside of our observable universe. Though, whether a light cone even has meaning before the inflationary epoch is a question of Grand Unification Theory, which is very much over my head.
As for further reading, Dodelson's "Modern Cosmology" is great if you're not afraid of learning the math behind General Relativity: http://amzn.com/0122191412 The text doesn't assume very much familiarity with GR, but it does assume enough mathematical sophistication that you can fill in any gaps in your knowledge on your own.
(And if you don't like the idea of learning about tensors, you're SOL. Sorry. There's a very low ceiling of how much one can know about cosmology without tensor calculus.)
I'm not fully understanding your next question. The origin of our light cone is here and now. Minkowski spacetime is an approximation that (almost) works in the absence of gravitating bodies. Once general relativity gets involved, certain features of Minkowski spaces start failing, like the fact that you can always reorient light cones so that they're parallel. This fails even in intergalactic space, where there is still a non-vanishing Weyl tensor effect due to Hubble Expansion. So, without getting too much into the detail, in the real world light cones are (forgive me) uncannily un-coney. Our past light "cone" collapses back in on itself in the distant past and converges on the Big Bang, but so do light cones everywhere in the Universe, even outside of our observable universe. Though, whether a light cone even has meaning before the inflationary epoch is a question of Grand Unification Theory, which is very much over my head.
As for further reading, Dodelson's "Modern Cosmology" is great if you're not afraid of learning the math behind General Relativity: http://amzn.com/0122191412 The text doesn't assume very much familiarity with GR, but it does assume enough mathematical sophistication that you can fill in any gaps in your knowledge on your own.
(And if you don't like the idea of learning about tensors, you're SOL. Sorry. There's a very low ceiling of how much one can know about cosmology without tensor calculus.)