Q Why do dogs lift their legs when they go to the bathroom?

A Dogs use their urine as a scent marker. You can think of it as a calling card, a signal letting other dogs know that they've been there. Many male dogs and some female dogs will ‘mark their territory' by urinating in many places. This establishes their scent on their territory, something that other dogs will readily pick up on.
It's not exactly certain why a male dog lifts his leg. By lifting his leg, the male dog is getting his urine on a vertical surface. One theory is that these vertical surfaces allow the scent of urine to last longer than horizontal surfaces do. Another theory is that this is more noticeable to other dogs because it's closer to nose level and thus less likely to be missed. I've also heard of a third theory which seems quite plausible. You may or may not have noticed that some male dogs seem to struggle a bit while lifting their leg, giving the appearance that they're actually trying to get the urine as high as possible. Some believe that this suggests that the higher the scent of the urine, the better. One reason for this might simply be that larger dogs can clearly get the scent of their urine higher than smaller dogs, thus the higher up the scent of the urine, the larger or perhaps more imposing the owner of that scent would appear to other dogs.