For what reason(s) is allolactose, instead of lactose, the "natural" inducer of lac operon repressor?
When lactose is present in a cell, some of it is enzymatically converted by $\beta$-galactosidase from the $\beta(1,4)$ linkage (typical of lactose) to the $\beta(1,6)$ glycosidic linkage (becoming allolactose). Allolactose and other analogues can then bind LacI to induce the appropriate conformational change and unbind the lac operator (one such review here).
Why does allolactose act as a signal of lactose presence instead of just lactose? From the perspective of the lac operon, there is no operational difference between lactose and allolactose -- either way one molecule binds LacI and cannot be metabolized while bound. The full paradox is explained by Edgel and summarized by well-known biochemistry textbook author and scientist, Larry Moran, on his personal blog. The short version is that neither allolactose nor lactose are the intended substrates for $\beta$-gal and instead a different sugar entirely is the "natural" inducer and substrate.