Image selection for Object detection
Do the images need to be the same/similar resolution as the camera I'll be using, for best results?
Technically yes, for best results, you should use as close as possible images that you train them on to predict, for example, HD camera and webcam images are quite different, but in real life, it is not usually possible, as it is costly. You don't need the same resolution images, but if you resize from 30 x 30 x 3 to 256 x 256 x 3  or 1024 x 768 x 3 it to 256 x 256 x 3 makes a difference in how the image looks after transformation, but  You can use different resolution images, but you need to resize them, using letterbox (padded image while preserving aspect ratio), or crop, or even in some cases just resizing without preserving aspect ratio gives good results.
It depends on a case by case.
Does the images all need to be the same resolution? Can I use a single image and just digitally zoom out on the image to give the illusion of different distances?
No, you don't have to have them in the same resolution. You can always resize images.
If the object is small in the image it could be possible, but remember it's better to have more unique data for generalization of the network. But for augmentation usually one of the augmentation techniques is zooming in (center crop and resize), and you could use zoom out as well.  There are other various techniques rotation, HSV transformations, shear, and many more you can use for augmentation.