Simplest remote router access is with a cloud router. A router with cloud features means you can use a phone app to configure the router from anywhere, although many normal routers allow remote access over port 8080. Not sure why you'd want to keep changing router settings though, as once set up it should be good.
Those security programs record images and videos as files. If you want to access the files over the internet the most straightforward way is through FTP, which many routers let you configure. The reason HTTP is generally not used is because your ISP will think you are hosting a server and want to charge you for it.