Account for the shorter C-O length in an ester compared with an anhydride?

A degree of cross-conjugation exists in the anhydride; this competition for electrons decreases the delocalization to each carbonyl O and gives the C-O sigma bond less double bond character.

Explain the significance of bond order. Can bond order be used for quantities comparisons of the strengths of chemical bonds?

Bond order is defined as half the difference between th electrons present in the bonding and the antibonding orbitals. Integral value of 1, 2, or 3 corresponds to single, double or triple bonds respectively as studied in classical concept. The bond order may be taken as an approximate measure of the bond strength. As the bond order increases bond length decreases and bond strength increases.

Why is dimethylformamide (DMF) a useful solvent?

DMF, HCONMe2 , is an aproric solvent (has no H for H-bonding) with a moderately high dielectric constant and a high dipole moment. Thus, although it dissolves many ionic compounds, DMF does not do so by forming  H-bonds with the anions. The salts are dissolved, chiefly through solvation of the cation by attraction to the end  of the C-O dipole. The + end of the dipole is shielded within the molecule and can solvate the anion very weakly, if at all.