Being totally indestructible and healing from damage that's somehow inflicted anyway is part of the basic IoP package.
This is true. Also the least of the TARDIS's abilities.
Does standing in it actually make you indestructible or does it just put an indestructible thing between you and the outside world? If the latter, IoP includes that too.
Technically? The second. For practical purposes since it's a building that can move anywhere and you can do anything from inside, and nobody can get in without permission? The first. Also, see below.
As for why to stat it, because if you're actually going to use the Doctor in a game he needs to pay for the abilities he has.
But that's the thing: Beyond getting him on scene and occasionally off again as an escape route, the TARDIS's potential capabilities are basicaly never utilized by the Doctor in any way. When and if they are it's because he worked out some way to do so that took as much time as any of his other technobabble solutions (ie: it's a use of Thaumaturgy).
The TARDIS is not part of the DFRPG's basic premise.
But it is a part of Dr. Who's. Indeed, a more major art than any city, and a very similar one in that it's only ever relevant to the plot very indirectly. Threatening the TARDIS happens, but in the same way that threatening the entire city happens in the Dresden Files, and the TARDIS enables the basic premises of a Dr. Who game the way a city does for DFRPG.
I'm gonna be honest, this whole plot device thing always kinda makes me groan. It's as if people are so impressed by something fictional that they want it to be awesome in a way that mere mortals cannot quantify. Even if it clearly isn't.
That's not at all how I use the term. I use 'plot device' specifically to refer to items that have widely varying capabilities that make no sense logically and work entirely based on what the plot demands, and are thus impossible to stat remotely accurately since, for any given episode, your stats are likely to make the whole thing make no sense.
The TARDIS fits this definition perfectly.