I guess it shouldn't: that is one of the controls usually included in this type of experiments, the reverse labelling, to show that any effect observed is not caused by the labelling itself.
Should not if done correctly. The forward/reverse labeling strategy is helpful as it a) provides biological replication and b) allows you to discriminate post-cellculture background from actual sample proteins, as the former should be strongly enriched on the light side in both experiments.
Some of the small detail on this may lie in detecting Proline-to-Arginine conversion - this is one of the known artifacts of SILAC labeling which can be circumvented by adjusting the Arginine concentration. Should be straightforward to find in the literature.
Isotope scrambling is a common problem for aminoacid-specific isotope labelling, and it could indeed cause your SILAC to be more heterogenous than intended. This is a common problem in biological NMR of large proteins, and I recommend you the following review for more information, with emphasis on figure 3 : 10.1007/s10858-017-0156-z