A:AnswerYes, you can just hook up your Cable Box using an HDMI cable.
Try the Power On feature, go Home > Settings > Display & Sounds > Power On and select Last Input.
A:AnswerIf all the HDMI inputs are showing No Signal, even if you connect other HDMI multimedia devices directly to the TV; please call 1 (855) 527-2411 available from 8 am to 9 pm CT everyday.
A:AnswerYou can connect your Cable Box to the TV using an HDMI cable for best video quality. Also you can connect your Roku device at any time.
David M. - Toshiba Support
A:AnswerThe TV should stay on as long as a video is playing on the screen.
For Toshiba TV product support, please call 1-855-527-2411 available everyday from 8am - 9pm CT.