I’m not sure if i get you right, but… HDMI is kind of an one way street. You can’t send any signal / data from your TV to the laptop that way. So you can’t control your laptop through the TV remote. If you hook up your TV through HDMI it becomes a stupid monitor. My best advice would be to get a small wireless keyboard/touchpad combo to connect to the laptop. Maybe a remote control you connect to the laptop will also do.
I don’t think that will work. Even a small jpg with under 1mb has to much data to get in one QR code. Not even considering multiple images. You probable can get one small svg icon into a QR code… but what’s the use case anyway?