Can an AND gate be used to toggle an HDMI connection?

Absolutely not.

First: HDMI uses TMDS signaling. This is not compatible with TTL logic gates; it's a current-mode differential protocol, very different from the single-ended outputs used in TTL logic.

Second: The TMDS signals used by HDMI run at over 1 GHz. This is far faster than the 74LS parts you're looking at can handle; indeed, I don't think it's compatible with through-hole parts at all. (The pins alone will have too much inductance for this fast of a signal.)

Consider using a dedicated HDMI switching part, such as the TI TMDS261B ("2 to 1 HDMI switch") or the Fairchild FDHDMI08 ("Wide-Bandwidth Differential Signaling HDMI Switch").

No, the signal frequency is way too high for typical AND gates. The signalling is also differential TMDS.

You may be able to apply your AND gate to the HPD (Hot Plug Detect) signal. You can then emulate unpluggnig the HDMI connector. Most sources will then stop transmitting, which is what you want.

As you would need dedicated HDMI switching chips to do the switching, which might be difficult to source and a PCB with the right footprint, etc. Allow me to make an alternative suggestion which might be easier for a one-off project:

Get a cheap, ready-made HDMI switch. For example, this 5-way HDMI selector on eBay has an infrared remote control. You could use the microcontroller to generate the infrared signals, or use a contact closure to press the button(s) on the provided remote control.

Just a thought. Maybe it helps.