I made an App Preview video only for the iPad resolution and uploaded it to iTunes Connect. If I don't upload a video for the other iPhone resolutions, will the iPad video by shown on the App Store when an iPhone user comes to the store page?
No, the iPad 'App Preview' video will only be visible for users who are viewing your app page from an iPad.
For iPhone users you would ideally upload three video at the following resolutions: 640 x 1136 (iPhone 5), 750 x 1334 (iPhone 6) or 1080 x 1920 (iPhone 6 plus). If however you decide to only upload one video (say 750 x 1334) larger iPhones will fallback to smaller videos if the larger preview video is not present.
N.B. These resolutions are for portrait mode.