So the container has its own file system and by default /storage inside the container is mapped to /mnt/user of your unRAID server. Also, the web interface you are accessing when opening the link under the Docker page is the X11 Tixiati application running inside the container. So in this context, yes there are some limitations (it's like a remote desktop). However, you can use the Clipboard from the side panel to share text between the host and container.
For the other point you mentioned: Tixati also offers a native web interface that you need to activate. To access this native web interface, you need to edit the container's settings and add a new "Port": this will setup a port mapping between unRAID and the container. By default, the port is 8888. So you would have to set the "Container Port" and "Host Port" to 8888. Once done, you can access this via <unraid ip>:8888.