You can output the same signal to analogue and digital outputs and measure the latency (from a PC. Don't think any consoles support dual display output). There shouldn't be any latency on the digital out so the end result will be latency of the digital input on your digital TV vs analogue input on analogue CRT (effectively zero).Then you only can test the input lag on the analog inputs, not on the HDMI input.
Yes. There's a widget you can buy that shows inputs, allowing measure of round-trip latency from game input to display output when used with a camera and frame-counted. If you already know the latency of the display, you can determine game latency, but you can't really get the display latency from using this controller without some sort of test app where you know the input latency. I suppose a known quantity like maybe a fighter with 1/60th latency could be used to count lag until output.IIRC Digital Foundry used a device that was using LEDs to show the exact moment when a controller button was pressed and then slow motion video was analyzed to determine the latency. Although they've used it on games only and not on the TV set itself.