This would be a good xkcd what-if, right up Randall's alley.

If you're talking about Earth-like atmospheric density, I can tell you that that is a lot of mass.

So first off, if it's not rotating, it's gonna collapse into the Sun. And I'd guess, without looking at the numbers, that it's probably gonna be some combination of processes that are gonna ultimately result in a black hole where the Sun is now.

If it's rotating, then there's a problem. You can't just have air staying in orbit in arbitrary orbits, because it'll collide. So I guess that you'd have to have a flat ring of gas, like Saturn's rings. Basically, it'd have an orbit of zero eccentricity.

The planets don't have an orbit of zero eccentricity, so they'd be smacking into air constantly. Aside from the effects of all of that air being captured by the planets themselves and adding to their mass, I'd guess that it'd tend to force them towards a zero eccentricity orbit.

If you have that gas ring, there may be more-dramatic near-term effects, but I would guess, again without looking at the numbers, that the planets would accrue it over time, and it'd have drastic effects on the planets; a lot of mass showing up, to say nothing of the chemical effects of oxygen and the thermal effects of colliding with the air.

It'd also add a lot of mass with rotational energy at orbital speed, so I assume that it'd make it easier for the Sun to capture stuff passing through that gas disc.

I don't know what effects would be sooner or dominate, but I think that it'd probably be pretty exciting in more unpleasant ways than the flying airplanes between planets thing that you mentioned.

EDIT: Hah, did this before reading mindbleach's comment, and he hit some of the same points.