Will my custom logo overlay & logo icon on the web player be displayed on OTT/ConnectedTV as well?

The player customization of the web-player happens player-side. Roku, FireTV and all other OTT/ConnectedTV platforms are using their own native players. None of them have the capability to display the logo overlay or logo icon from your customized web-player. The only way how to add a logo overlay on OTT/ConnectedTV platforms is to have a watermark directly rendered onto your videos during the video editing process, or by using the "Lightcast Watermark Feature" (included in larger packages, see www.lightcast.com/packages ). 

Adding overlays on web-players and rendering a watermark onto the video-file itself are two entirely different technologies and solutions, serving different purposes. One difference is that OTT/ConnectedTV Apps will display your watermark if directly rendered onto your video files.