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We have applied the image subtraction method of Alard and Lupton to the extensive M3 data set previously analyzed by Corwin and Carney using DAOPHOT and ALLSTAR. This new analysis has produced light curves and periods for 15 variables not found in the previous study but already known to be variables, and it has also resulted in improved periods for several other variables. The additional variables recovered with the image subtraction analysis are in the very central region of M3, where crowding is severe and the photometry was not of sufficient quality that it could be put on the standard system. The present study brings to 222 the total number of RR Lyrae variables in Corwin and Carney's M3 data set for which light curves and periods are available. Among them we have identified three new candidate double-mode pulsating variables (V13, V200, and V251), reported here for the first time. This brings to eight the total number of double-mode RR Lyrae (RRd's) identified in M3. Of the newly discovered RRd's V13 is unusual in that it has the fundamental as the dominant pulsation mode. M3 is unique among the globular clusters in having RRd variables with a dominant fundamental mode. Two of the new candidate RRd's (V13 and V200) have period ratios as low as 0.738-0.739. They lie well separated from all previously known double-mode variable stars in the Petersen diagram in positions implying a large spread in mass and/or, less likely, in heavy element mass fraction, among the M3 horizontal-branch (HB) stars. We explore mass transfer and helium enhancement as possible explanations for the apparent spread in HB masses. We also note that the masses derived from the double-mode analyses now favor little mass loss on the red giant branch. We find that V200 has changed its dominant pulsation mode from fundamental to first overtone, while V251 has changed its dominant mode from first overtone to fundamental in the interval 1992 to 1993. Together with M3-V166 this is the first time that double-mode variables are observed to switch their dominant pulsation modes while remaining RRd's. The phenomenon is found to occur in a 1 yr time span, thus suggesting that these stars are undergoing a rapid evolutionary phase and that both redward and blueward evolution may take place among the HB stars in the Oosterhoff type I cluster M3. The unusual behavior of the M3 RRd's is discussed in detail and compared with that of the double-mode RR Lyrae identified so far in globular clusters and in the field of our and other Local Group galaxies. We find a lack of correlation between the presence of RRd variables and any of the cluster structural parameters.