A MOV from an iPhone or a Mac will not always play, or upload, on other devices and websites. An MP4 almost always will.
MOV is Apple's container for video, and a MOV from a recent iPhone often uses HEVC video, which many Windows and Android devices and websites cannot play. An MP4 with H.264 video is the format nearly everything accepts.
Converting means encoding the video again, so it takes time in the browser and costs a small amount of quality. Convert from the original file rather than from a copy, and keep the original.
If the video is large, you may also want to shrink it. The Compress video tool does that and saves an MP4 as well.
A little. Best quality keeps the loss very small.
Yes.
No. The conversion happens on your device.