What are the generators and relations of the conformal cobordism category?

Question

According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming from a TQFT background, I find this a very sensible and clear approach to CFT. However, I haven't found any place so far where this approach is really followed through in a straight-forward manner. In particular, I haven't found an explicit presentation of the conformal cobordism category in terms of parametrizations of the corresponding conformal moduli spaces.

Question: Is there a presentation of the conformal cobordism category in terms of generators and relations, labelled by a finite number of real parameters? If not, I'd be happy with at least some partial insights, like the dimensions of the relevant moduli spaces, or the generators alone.

Here my partial thoughts/guesses on the subject, please comment and correct:

1. It seems pretty reasonable to assume that it suffices to consider all the conformal pairs of pants, conformal annuli, and conformal disks as generators (together with a cup and cap if we distinguish input and outputs), and likewise all conformal versions of the 2d TQFT relations as relations.
2. A genus-$g$ surface with $n$ punctures for high enough $g$ and $n$ (such that the conformal automorphisms vanish) as a manifold with boundary has a moduli space of dimension $6g-6+3n$. However, conformal cobordisms also come with an embedding of a standard annulus into the vicinity of each boundary circle which yields a moduli space of dimension $6g-6+4n$, with one more parameter per puncture corresponding to rotating the embedding of the annulus.
3. An annulus has a 2-dimensional moduli space, corresponding to the ratio of radius and width, and the rotation between the two embedded boundary annuli.
4. The pair of pants has a $6$-dimensional moduli space. Modulo gluing annuli to the three boundary circles, the moduli space is $0$-dimensional. Maybe it is possible define a single pair of pants where "the surface between the punctures is infinitely thin", from which any other pair of pants can be obtained by gluing annuli?

If there exists an answer, I also have a few bonus questions:

1. Are there examples for CFTs for which one can really spell out the tensors associated to different conformal cobordisms?
2. What is the interpretation of the (chiral) vertex (operator) algebra, operator product expansion, and Virasoro algebra in terms of the Segal definition?
3. In particular, it looks like the vertex operator algebra is the tensor associated to a conformal pair of pants, and it is tempting to think that the parameter $z$ therein parametrizes different pairs of pants in the conformal moduli space. Is that a correct interpretation, and if so, where are the other $4$ parameters?
4. Is there a way to see directly from (some part of) the generators and relations, that any functor to a finite-dimensional vector space must necessarily be topological?