It's been a very long time since I've read this paper, and I haven't been able to find a copy online, so my apologies in advance if what I'm about to write is nonsense.
Think in terms of projective coordinates (i.e. I won't say "cone" any more). I think the lemma you describe also holds for surfaces $S$ on the boundary of the face in question. (If this is not the case, then never mind, I'll delete this answer.) That is, we can take $[S]$ to be a vertex of the face, and we know that $t[S] + u\alpha$ is representable by a nonsingular 1-form whenever $\alpha$ is, for $t \ge 0$ and $u>0$. In other words, starting at a representable point in the interior of the face, we can move a arbitrary distance toward any of the corners of the boundary of the face, so long as we stay in the interior. Since the interior of the face is the open convex hull of the corners, we can reach any point in the interior this way by letting $S$ run through the corners of the face.
(EDIT: As Agol points out in a comment below, instead of "corners" I should say rational points in the boundary of the face.)
A good place to start is in low dimensions and there a good introduction can be found in
J. Kock, 2003, Frobenius Algebras and 2-D Topological Quantum Field Theories, number 59 in London Mathematical Society Student Texts, Cambridge U.P., Cambridge.
There are also old notes of Quinn that take a very neat homotopy theoretic approach to some of the problems:
F. Quinn, 1995, Lectures on axiomatic topological quantum field theory, in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory, volume 1 of IAS/Park City Math- ematics Series, AMS/IAS,.
These do not really get near the physics but you seem to indicate that that is not the direction you want to go in.
I like the links with higher category theory. I realise that this is not everyone's `cup of tea' but it does have some useful insights. I wrote a set of notes for a workshop in Lisbon in 2011 which contain a lot that might be useful (or might not!). They are available at
http://ncatlab.org/nlab/files/HQFT-XMenagerie.pdf
My advice would be to raid the net getting this sort of resource (storing it on your hard disc rather than printing it all out!), then as you start working your way through some of the stuff you have found there will be explanations available ready at hand. Start with the main ideas and `back fill', i.e. don't try to learn everything you might need before you start. If when reading some source material an idea that you are not happy with comes up, search it out then, just enough to make progress beyond that point easy. (Of course this is how one progresses through lots of areas of maths so ....)
(Those notes of mine exist in several different forms and lengths, so in a longer version some idea may be more developed.... so ask!)
.... and don't forget the summaries in the n-Lab can be a very useful place to start a search.
(I had at the back of my mind just now a reference to a seminar that I had some notes on .... but no idea of the source. A little search found me:
http://ncatlab.org/nlab/show/UC+Riverside+Seminar+on+Cobordism+and+Topological+Field+Theories
... One other thing, there are lectures by Lurie on this stuff on YouTube:
https://www.youtube.com/watch?v=Bo8GNfN-Xn4
that are well worth watching.)
Best Answer
For the sake having an official answer, as opposed to a comment-as-answer, I'll second Mark Grant's suggestion of Rolfsen's "Knots and Links". It was the first book I read on 3-manifold topology, and I enjoyed it very much.