Can I hook up my Apple TV to my Xbox One?

You can connect the Apple Tv to the HDMI Input on the Xbox One, and when you turn it on its image should appear on the Xbox's main screen for you to select.

The slowness or lag should only affect really fast paced games. Movies and regular videos should not have any noticeable difference.

no it mean that an xbox is sort of like a computer when it's running it's using power from your ac mains

this produce heat and cost on the powerbill

if running an xbox would be a fire hazard then Microsoft would have a serious problem, but connecting an appletv to an xbox and having both running to watch appletv shows on a tv would cost the power that the appletv use plus the power that the xbox use

a "I heard you like cars so I installed a car inside your car so you can drive your car while you're driving your car" thing

No it's not. No one asked if you thought their power bill would go up, they asked for help setting their system up. That's how I got here. Not because my power bill is high now after installing Apple TV through XBOne. I wouldn't have to run my Apple TV if XBOne had a DIRECTV app. I also wouldn't have to run my XBOne if Apple TV, like the iPhone, had an Amazon Prime app. So now I run both through my Plasma sitting in a room glowing the sweet light of incandescent bulbs.

I haven't tried but I doubt it. You would want to connect the ATV via HDMI directly to the tv/receiver...

Out of curiosity, what is the desired outcome for this type of setup?

It does work just have to set up as instructed in one Tv selection. No problem.

it's just that the xbox use a lot more power then appletv both in standby and operation

What does "use a lot more power" mean? Is it a fire hazard or something?