Does the rear tyre carry more weight than the front tyre? As I'm riding my bike, it looks as if the rear tyre is carrying more weight than the front tyre: The rear tyre looks more squished downwards, even though both are equally well-inflated. 

This I think has been true for every bike I've had. 

Am I just imagining it or is this generally so?

Might this also explain why the rear tyre is more susceptible to wear and tear than the front?
Generally, the weight distribution of a rider on a bicycle is something like 40 - 60 rather than 50-50 (front to back). This contributes to increased rear tire wear (along with rear wheel drive) causing front tires to sometimes last three times as long as rear tires. Skid stops (such as on a fixie) can also increase rear tire wear relative to normal stops.

An aside: You don't need to run your tires at equal pressure - in fact, in most cases, the rear tire is about 10% higher than the front tire. This will make it deflect a bit less.