Digital - we got one pretty cheap on Amazon.
Once per week. Daily weighings will drive you crazy. It's overkill, and it's inaccurate anyway. Weekly weighings (make sure it's the same day, and more or less the same time every week) make more sense as that takes advantage of the fact that weight fluctuates not only based on food and drink consumed, and exercise, but also because of the weather (how much you sweat), when you last had a bowel movement, and the contents of your meals for the past week or so, even if the calories were the same. E. g. 100 calories of skinless chicken breast is going to affect you differently from 100 calories of Oreos.
I don't think the weight estimate on the treadmill matters vis a vis how hard you work on the treadmill. What it does, instead, is it changes the expended calorie estimate. Heavier people expend more calories for identical exercise because they are hauling around more weight no matter what they do.