Here is an install video from Tushy showing all the steps. https://youtu.be/jZKxmk_dIVg

I can't speak for all bidets. I got one from Tushy and it came with a short metal hose and T connector.

It would be no problem to hook that up to the water connectors you show there. Step 1) turn off the water supply by turning the cold water knob down in the picture. Step 2) disconnect the water hose and install the T connector where the hose was. Step 3) connect the sink hose and the bidet hose to the t connector. Step 4) install the bidet and connect the other end of the bidet hose to the bidet

At this point it should be all set up with child water going to the bidet. If you are thinking about hooking up the hot water instead you should consider some things first. How long does it take your sink to get hot water flowing when you turn on the hot water only? In my house it can take a minute which is longer than you'd use the bidet, so no point in plumbing hot water. The other thing to consider is how hot is your hot water when it gets flowing? If it's hot enough to burn think twice before pluming that into the bidet. You don't want to burn your ass.

The picture isn't great but it looks like a messy spackle/mud job to me

That was my first thought. Just being manipulative and seizing the opportunity. Who hides a ring in a mattress?