Chrome crashes but stays displayed on external screen and I can’t remove it

All I can really suggest is that you purchase a Mini DisplayPort to VGA Adapter, along with a VGA cable (D-Sub cable), and use that with your MacBook Pro.

(the blue socket beneath the DVI socket is the VGA socket.)

When connecting an external display to a MacBook Pro (MBP), sometimes you'll get lucky, and both electronic devices will recognize each other and adjust their own settings properly, and the laptop will be displayed on the TV. However, this doesn't always happen. When it doesn't...

First, always check and make sure the external display (TV in your case) is powered on and all its cables and connections are properly seated. In order for the TV to sync, a few things need to happen, the MBP has to recognize that a TV is connected (which you say is happening), the TV has to recognize that a video source is present (unclear if this is happening), the TV has to be set to the proper Input or Source (Input numbers on the back and sides of TV don't always correlate to the Input number in the TV Menu), and the format of the video coming from the laptop must be supported by the television.

It sounds like the laptop is doing everything it should, so focus your attention on the TV. Cycle through all the different inputs in the TV. Try different HDMI ports. If that doesn't work, it could actually be the Mini DP to HDMI adapter. Try and connect the laptop and adapter to another HDMI device. If it still doesn't work the another device, then the adapter may be bad. If it does work with another device, then either the TV has an issue with HDMI inputs, or the settings in the TV are preventing it from working properly. Possibly try and reset the TV back to its default settings.