tile at the door would be my choice. the style/design would depend on the rest of the room... a photo of the whole room would help with suggestions.

with the overhang of the upper, I'd choose something light to help bounce more natural light around the room.

There’s another view! Thanks- definitely going light w carpet color to brighten things up

Instead of a mat, a dense low pile bound carpet in the same shade as the main piece is my contribution. Often, manufacturers offer different grades in matching colors. Or possibly rotate three bound pieces of the main carpet material to allow for shampooing the protective piece. You'll likely end up with the extra material for free.

Doesn't make sense to put wall to wall down and then place a large area rug over it. With kids I would do some type of LVP for the whole room and then use my area rugs with good padding under it.

When you pull up that wall to wall, pay close attention to the dirt that you are going to find in and under the carpet, even if you were a great housekeeper and vacuumed more often than the average person.

I trully have no idea whay anyone chooses wall to wall carpet even in bedrooms there are so mny fabulous choices for flooring many even cheaper than carpet . So can't help there You need much better fuller drapes and again not a great choice with traffic in and out all the time . A nice LVP waterproof looks great in a light color and washable area rugs IMO best choice If you insist on carpet then everyone uses an out door mat to wipe feet and remove shoes .