Your tablet does not need to be on for this device to work. The tablet will be needed to set the speaker up (connect it to WiFi, link subscriptions accounts like Pandora or Netflix, link other google devices to the speaker, etc) but once the speaker is connected to WiFi, no tablet is necessary to play music, answer questions, or stream video.
Your smart TV will work with the Google Home Mini if your TV has embedded Chromecast. If so, link the two to the same google account, share the information on the Google Home app, and you will be able to stream using voice command. If your TV does not have Chromecast, you will need to purchase a Google Chromecast device.