If you're only delivering to the HTC One mini, then yes, screen resolution overrides source resolution, as you put it. Your assumptions are correct about 1080p video either having to be scaled or cropped to fit the display, and you're right, whether the source media is scaled or cropped does depend on the playback application.
As a general rule, you should try to match your final output to the format upon which it will be viewed. In your case, because the HTC One mini is only capable of displaying 1280x720 (roughly 900,000 pixels), there's no need to send it 1920*1080 (about 2 million pixels). Doing so is certainly a waste of bandwidth, and arguably a waste of CPU cycles and battery life.
But to be clear, this advice only applies when you know the video will never be displayed on any other device. If your final output is 1280x720, then anything with a better display will have to scale the image UP, making it blurry or blocky. By limiting your resolution to the HTC One, you'll notice sub-optimal performance on HD-TVs, for instance, or perhaps next year's model phone.
On a screen that size, can anyone actually tell the difference between
a 1280 x 720 HD and a 1920 x 7080 Full HD movie?
Probably not. It depends on the quality of the phone's onboard scaling vs the quality of whatever application you'd use to scale from 1920x1080 to 1280x720.
Further to this, what's the difference between a screen resolution of
1280 x 720 on a 4.5" screen than to a 50" screen? The pixels are
bigger on a 50" screen so looks 'worse' (Or pixels are smaller on a
4.5" so it looks 'better')?
Yes. If you have the same number of pixels on a bigger screen, the pixels themselves will just have to be bigger. But you have to remember that a 50" screen isn't meant to be viewed at arm's length like your phone.