Is there any way to export/download a video via REST API, by specifying the start and end time, as well as the camera from which the video is to be exported/downloaded?
No, unfortunately this is currently not implemented in Rest API.
For now you will have to use either MIP SDK or Mobile SDK..
https://doc.developer.milestonesys.com/mipsdkmobile/index.html?base=reference/protocols/mobile_command.html&tree=tree_1.html [CreateExportDownloadLink]