Problem : Which of the following is the best SN1/E1 nucleophile and base?

Molecules B and C are both strong bases and good nucleophiles. They will promote SN2 and E2 reactions, not SN1 and E1 reactions. Molecule C is a polar, protic solvent and will accelerate the two unimolecular reactions.

Problem : Which of the following two molecules is likely to have a greater proportion of E1 products?

Molecule A is a better candidate for elimination. It's α-carbon is more hindered, and its double bonds will be more stable as per Saytzeff's rule.