What do you call that little area just inside the front door of some houses?

In the UK this would most commonly be called a hall assuming it has doors, arches or similar leading into other rooms. A hall is normally a long, often narrow, room with several rooms opening off it (corridor is rarely used inside the home) but the minimal hall is probably the one I had in my old house: about 1 metre square, with the front door living room door and stairs leading from it.

As to whether it's considered a room, that's variable. When thinking about decorating, for example, yes - "which room are you going to decorate next?", "The hall". But in terms of counting rooms to describe the size of the house, generally a hall isn't a room. Wider examples may be an exception to this - some are big enough to use a corner as an office or seating area.

It's called a porch. After the porch is normally a hallway leading to the rest of the house and a stairway leading upstairs.

Coat room, if it has a closet. Or no name at all, just “by the front door”. Atrium is formal and correct but too solemn for a single room studio apartment.