So, the image above represents the resonance contributors of (E)-1-methoxybuta-1,3-diene. In another thread on this forum it was explained to me that because oxygen is so electronegative, it does not share electrons AT ALL and, thus, does not appreciably participate in resonance.
Short answer, no.
Question 1: Is it the conjugated diene that's allowing the methoxy group to be the partial positive end of the dipole? That seems fairly counterintuitive to me.
Yes, the oxygen can donate non-bonded electrons. What makes this counterintuitive is any thought that oxygen, as an electron withdrawing group, will not donate its electrons. Oxygen can and does share its electrons in many reactions, for example ketalization reactions.
Question 2: Can conjugated double bonds be thought of as bestowing an instantaneous double bond character to all C-C bonds spanning the conjugated chain? As in, should I picture it like this:
Although a conjugated double bond does confer some double bond character to the conjugating single bond, it would be an exaggeration to represent it as though it were a double bond. That is, it is not equivalent to the bonding of a benzene ring in which one might write it with the dashed single/double bonds.
Diels-Alder reactions of this type show a preference in which one can use these resonance structures to predict the orientation of the major product. However, if this is considered a concerted electrocyclic reaction, it is difficult to justify its mechanistic implications. For me, I use these structures as a guide to predict the products rather than the mechanism of the reaction.