Why are there many examples of 2-methylquinolines being treated with a base then an electrophile but none of the corresponding reaction with 3-methylquinolines? Does this have to do with the charge not being stabilised for the 3-methylquinoline?
Yes, in contrast to 3-methylquinoline, the methyl group of 2-methylquinoline (1) is easily deprotonated by potassium ethoxide.
The anion has been examined by NMR spectroscopy (DOI), but its use in synthesis was reported long before. There's an article (in German) in Chem. Ber., 1909, 42, 1140 (DOI) where the reaction of 2-methylquinoline with diethyloxalate is described.