Frontier orbital analysis of zinc-catalysed carbonyl-ene reaction Q1. Is the $\sigma_\ce{C-H}$ bond a donor in this reaction? If not, then can we take the alternate combination (HOMO of alkene, LUMO of the $\ce{C-H}$ bond and, HOMO of the carbonyl)?
Q2. We can also draw the LUMO of the carbonyl group in the other possible manner(simply flip the shaded lobe on each atom, hence keeping the node). The reaction would then have to go through an antarafacial manner.
Why do we draw the LUMO in this manner?
First of all, a molecule only has one HOMO and LUMO. Don't confuse MOs with localized orbitals from VB theory.
You can basically approximate this mechanism with four MOs:

LUMO+1: Mainly $\pi^*_{C=C}$ with contributions from $\sigma^*_{C-H}$
LUMO: Mainly $\pi^*_{C=O}$
HOMO: Mainly $\pi_{C=C}$ with contributions from $\sigma_{C-H}$
HOMO-1: Mainly $p_O$ with contributions from $\sigma$-orbitals on the carbonyl carbon. We can regard this MO as one of the lone pairs of the oxygen.

The corresponding interactions will roughly look like this:

Hence, the HOMO - LUMO interaction explains the C-C bond formation, while the HOMO-1 - LUMO+1 interaction explains C-H bond breaking and O-H bond formation. Both HOMO and LUMO+1 also explain the shift of the double bond.
Now to your question:

Q1: Yes, $\sigma_{C-H}$ is part of HOMO, but has a lower contribution compared to $\pi_{C=C}$. So I would mainly call the C=C bond as donor. As you can see $\pi_{C=O}$ is not part of the FMOs, so your proposal makes little sense. It mainly contributes to more stable occupied MOs.
Q2: If I understand correctly, you propose an antarafacial [2+2], which is in principle symmetry allowed, but would lead to a very strained transition state with a high barrier. You are free to swap the sings on MOs, but as soon as interact with another MO, you take the one that leads to stabilization of course.

Edit: As reference, I got the FMOs from a HF/6-31G calculation on a slightly simplified system: