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I think the main reason it appears to rise at different spots each evening is because it is around 50 minutes later each time? The orbit appears at an angle to our horizon and so when it rises, it will appear to have moved around a bit.
This is probably easier to check when setting because it will be easier to grasp the concept. I expect if you watch where it sets one night and the next day note where it is about the same time as it set the previous day, and drop a line straight down to the horizon, it should be the same place? Well, that's my theory. The Sun does a similar thing, but the time difference is only 2 minutes (roughly) and so the difference is pretty small, making it appear to "walk around the horizon" during the year.
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