Which suit is it?
The Marvel Movies Wiki has a list of all the Iron Man suits, along with helpful pictures. As of Iron Man 3, there are 42 in total, and we can compare these pictures to the scene to work out which suit Tony is using.
For reference, here’s a picture of the suit in the panic attack scene:

So what’s notable about this suit?
It features the traditional red and gold livery. Surprisingly few of his post-New York suits featured this colour scheme, so our choice is narrowed down to a dozen suits almost immediately: 3, 4, 6, 7, 8, 9, 10, 11, 17, 33, 35, 42
The arc reactor is circular. Suits 6, 33 and 35 can be dismissed because they have triangular or rectangular chest pieces.
It’s a traditional design. There aren’t any zany additions, like claws or extra repulsers. A further three suits (11, 17 and 42) can be dismissed because they look very different to this suit.
It’s fairly battered. All of Tony’s new suits are pristine: he built them, and then they sat in his basement. This suit has seen a lot of action. The last suit we saw him using was the Mark 7, at the end of Avengers, so we can lean towards ruling out 8, 9 and 10.
By my counting, that leaves us with suits 3, 4 and 7. There are several reasons that would lead us to believe that this is probably the Mark 7:
- The grey-coloured plates on his shins. They don’t appear on 3 or 4, just the Mark 7.
- Those brackets over the shoulder. Another design feature unique to the Mark 7.
- Only the Mark 7 seems to be in working condition. The Mark 3 was badly beaten up by the Iron Monger in Iron Man. Promo pictures show the suit as fairly damaged, and there’s no guarantee that it will still work. Meanwhile the Mark 4 suit got decapitated by flying chunks of the Mark 42 early in the movie. Probably only cosmetic, but also not something he’d take out the next day.
- Only the Mark 7 has been seen with this “easy access” routine. The Mark 3 is the first red-and-gold suit, and in Iron Man we see him getting into it with a team of robots. Similar for the Mark 4.
You can compare other details on the suits; the result is pretty clear.
So this suit is the Mark 7, last seen when Tony was using it in The Avengers.
(The Marvel Wiki seem to agree, but they don’t provide any evidence.)
Which suits can do this?
We see multiple suits do this at the end of Iron Man 3, but I’m not going to make a list of individual versions; most of them are on-screen for just a few seconds.
We know this functionality was introduced with the Mark 7: he has to use his moving archway to take off the Mark 6 (in Avengers), but we see him use the Mark 7 like this at least twice.
I think it’s reasonable to assume that every subsequent suit could do this, as it’s far more convenient than the alternative (having to rely on an assembly robot to take it on or off). The only exception might be the Mark 42, whose new modular design might make this behaviour impossible or impractical. But we see the Mark 42 do this at least once in Iron Man 3.
So I believe every suit from the Mark 7 onwards has this easy-access behaviour.