The motive $L$ is called Lefschetz because it is the cycle class of the point in ${\mathbb P}^1$, and so underlies (in a certain sense) the Lefschetz theorems about the cohomology of
projective varieties. To understand this better, you may want to read about how the hard Lefschetz theorem for varieties over finite fields follows from the Riemann hypothesis, as well as a discussion of Grothendieck's standard conjectures and how they relate to the Weil
conjectures.
The motvie $L^{-1}$, when converted into an $\ell$-adic Galois representation, is precisely the $\ell$-adic Tate module of the roots of unity. Tensoring by this Galois representation is traditionally called Tate twisting, and so the motive underlying this Galois representation is called the Tate motive.
One needs to have $L^{-1}$ at hand in order for the category to admit duals.
If one were working with just usual singular cohomology, this wouldn't be necessary; Poincare duality would pair $H^i$ with $H^{\text{top}-i}$ into $H^{\text{top}}$, which would be
isomorphic with ${\mathbb Q}$ via the fundamental class.
But motivically, if $X$ (smooth, connected, projective) has dimension $d$, so that the top dimension is $2d$, then $H^{\text{top}} = L^{\otimes d}$, and so $H^i$ and $H^{2d-i}$ pair into
$L^{\otimes d}$. To get a pairing into $\mathbb Q$ (the trivial 1-dim'l motive) we need
to be able to tensor by powers of $L^{-1}$. Traditionally tensoring by the $n$th tensor power of $L^{-1}$ is denoted $(n)$; so we find e.g. that
$H^i$ pairs with $H^{2d -i}(d)$ into ${\mathbb Q}$, and we have our duality.
You can see from the fact that cup product pairs cohomology into powers of $L$ that inverting $L$ is precisely what is needed in order to obtain duals.
Finally, one should think of $L$ as the fundamental class of a curve,
think of $L^{\otimes d}$ as the fundamental class of a smooth projective $d$-dimensional
manifold, and also become comfortable with Poincare duality and the Lefschetz theorems;
these are the basic ideas which will help give solid geometric sense to motivic constructions.
One reason is Givental's conjecture, which says that in the semisimple case, genus 0 GW invariants determine higher genus GW invariants. See this paper of Teleman, in which the conjecture is proved.
The theory of Frobenius manifolds in general is quite complicated. I guess semisimple Frobenius manifolds form a relatively tractable set of examples. Here are some basic references for the theory of semisimple Frobenius manifolds:
Manin: Frobenius manifolds, quantum cohomology, and moduli spaces
Dubrovin: Geometry of 2d topological field theories
Lee, Pandharipande: Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints (so far unpublished and incomplete; available at Pandharipande's webpage)
Semisimple Frobenius manifolds also arise in singularity theory, when studying for instance isolated hypersurface singularities (see Hertling's book Frobenius manifolds and moduli spaces for singularities; the three references above probably also talk about this), or in the physics terminology "Landau-Ginzburg (B-)models". I don't know whether Frobenius manifolds (in particular non-semisimple ones) arise more generally in singularity theory...? In any case, these Frobenius manifolds coming from singularity theory are supposed to be related to those coming from Gromov-Witten theory via mirror symmetry.*
Another comment: Quantum cohomology of, for example, $\mathbb{P}^n$ is semisimple. Then perhaps this makes quantum cohomology and GW theory of projective varieties more tractable, because of quantum Lefschetz ... but I don't really know anything about this. But very roughly speaking, I think this is the strategy of Givental in his proof of the "mirror conjecture" of Candelas et. al. regarding the genus 0 GW theory of the quintic 3-fold, though I might be wrong.
*Edit: For example, this paper of Etienne Mann seems to prove a mirror theorem relating the quantum cohomology Frobenius manifolds of (weighted) projective spaces and the Frobenius manifolds associated to the mirror Landau-Ginzburg B-models. As Arend mentions, germs of semisimple Frobenius manifolds are specified by a finite set of data, and I think the strategy of Mann's paper is to show that these data coincide for the two Frobenius manifolds.
Best Answer
I'm not sure what you mean by "every projective scheme should have a quantum cohomology structure". In the talk abstract that you link to, it does not say "projective scheme" but "smooth projective variety". I don't know whether the theory generalizes to non-smooth things or to things that are not varieties.
Quantum cohomology is a deformation of ordinary cohomology (or Chow ring if you like) of a smooth projective variety (or compact symplectic manifold). This structure comes from (genus 0) Gromov-Witten invariants. GW invariants are constructed using the ordinary cohomology of your variety/manifold and the ordinary cohomology of moduli spaces of stable maps and stable curves. I mostly work over $\mathbb{C}$ so I don't know too much about what I'm about to say, but if you're not working over $\mathbb{C}$, then ordinary cohomology doesn't make sense, but instead you can still work with things like $\ell$-adic cohomology or crystalline cohomology. This is what "motives" refers to. I guess Manin is saying that just as you can do cohomological (and Chow) Gromov-Witten invariants and quantum cohomology, you can also do the analogous things for motives. I suppose the resulting things would be called "motivic Gromov-Witten invariants" and "quantum motives".
I'm not sure whether it makes sense to ask about the behavior of quantum cohomology under monodromy. As I understand it, monodromy refers to using a connection (Gauss-Manin connection) to parallel transport (co)homology classes. You can view quantum cohomology as being simply ordinary cohomology except with coefficients in a Novikov ring and with a deformed cup product. Viewed as such, the monodromy of quantum cohomology should be the same as the monodromy of ordinary cohomology, because "quantum cohomology classes" are no different from ordinary cohomology classes.