I would have it painted the same color as the walls

For modern farmhouse, I’d stain it all very dark, like MinWax True black.

Yes, I want the TV there. I have kids. I am thinking to put drywall behind the TV and leave just the right space for it.

Or drywall the entire area and surface mount the TV. Then you could change TV's easier without being tied to the existing space.

Karoline, I like your idea about the drywall. I would use the furniture as is. Since it's a custom built-in piece I'm sure it is real wood. I think it looks nice and shouldn't be painted at all. Not everything in modern farmhouse style has to be white, it's the accessories you use that really matter. I like the doors as they are, but if you want to make them look more farmhouse, you could replace the glass with some type of screening, or even put some type of design on the glass. I like to look at some websites like the one below that have farmhouse type decor to get ideas.

https://www.countrydoor.com/c/furniture/living-room/cabinets/