Are headphones now bluetooth to connect to cellphones?

A few newer cellphones like my A52 still have the audio jack, but if he has an iPhone then it's only Bluetooth without an extra dongle/adapter.

There are headphones like the Bose QC35 II that can both use a cable and Bluetooth.

connect to the TV

I've seen the TVs at my clinic. Bluetooth won't work so you need one with an audio cable. Well, that's not entirely true. I have an MP3 player to play music without my phone, and I sometimes use a Bluetooth transmitter+receiver in place of the audio cable. But using a cable is probably simpler.

I think noise cancelling might be important to cancel the machine noise but I don't really know how it works.

Up to you but personally I recommend noise isolation. The way noise cancellation works is it has a small microphone to listen for low-frequency noise and emit a wave to cancel it out. That works well for constant noise, like humming of air conditioning.

My experience is at the center there are a lot of alarms and a lot of people talking loudly, and then also people watching TVs without headphones. Noise cancelling isn't really good for cancelling that sort of high frequency noise. See article for more information.

These are the headphones I use, personally.