Well, given a sphere of air stretching out to Neptune would only be around 6.5x its Schwarzschild radius in diameter, I imagine it would very quickly collapse into a black hole. There definitely won't be enough time to fly airships around or hear the sound from the Sun before the planets are shredded by the air (which is moving at roughly Mach 80 in the case of Earth), even if they didn't fall into the new black hole.
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.
Now I'm curious if everything orbiting in space would "burn up" from the sudden friction like when an orbiting object comes into contact with an atmosphere.
Well, it wouldn't change the gravitational field of any given planet so the atmospheres around planets would exist still but they might become more air saturated. The other thing to consider is that since this air would be so far away from the gravitational fields of many planets that it would actually get pulled into the sun over time, increasing the mass of the Sun and further accelerating the collapse of the solar system. By how much I could not tell you, but eventually the extra mass being dumped into the vacuum of space around our solar system would increase the mass of our star.
that's roughly a protoplanetary disk, what it's like before new planets form! jupiter's mass is 95% hydrogen-helium atmosphere with a core of ice and metals. because the solar system already has planets, though, we'd expect the ones currently here to "sweep" up the gas along their orbits in a fairly timely manner (jupiter's year is 12 earth-years). if the gas is spinning at about planetary speeds, most of the gas will gradually be pulled out of solar orbit and into planetary atmosphere, so basically all the planets will become gas giants (or possibly new dwarf stars! i haven't done the math). depending on how far out your protoplanetary disk goes you might need new planets form in the outer regions, past the regions where existing planets have formed. expect orbits to get really fucked up due to new gravity dynamics.
A: yes, you should have enough air pressure to leave orbit in a plane, which is great to flee the planet because the new solar atmosphere will immediately plunge earth into an ice age for a few years as it first blocks most of fhe solar radiation, and then gradually heats up. the solar atmosphere wind dynamics will also fuck you up in a way that the dying Earth's scientific computing will struggle to help with, especially when all satellites burn up in air and fall to the surface
B: (1 AU / the speed of sound) = 5000 days. the distant, ominous howl of a vengeful sun will be the least of your concerns as the nations of earth turn violent in a desperate attempt to save some last trace of human civilization
C: a megacyclone, yes! a beautiful spiral as each planet sweeps up the protoplanetary gases to swell to gas-giant size or dwarf star status. perhaps some day the light of five suns may shine on Earth again, but all traces of what once was will be sunk under five thousand kilometers of a liquid nitrogen sea. in one possible delusion, the billion-year descendants of the last earthlings will live in orbit of a still-molten rock so far past the kuiper belt that the five stars of Sol look like a bright-hot constellation instead of the core of its orbit
D: at some point, yes! for a brief moment the pressure will be unchanged as the solar atmosphere orbits the sun instead of pressing on the earth. perhaps some would be wise enough to realize that the sudden dimming of the sun - eight minutes as it turns to a dull, occluded, sky-colored haze, as if it had just melted away - foretold an imminent disaster. those bright minds would struggle to be heard amid a hundred other apocalyptic visions, because who would believe the sudden, impossible appearance of so much mass? but maybe some would have the sway and the speed to take their calculations to an appropriately powerful spaceplane, or a convenient tanker aircraft, and set off before the earth's mass doubles in a year, then doubles again, like a snowball of oxygen rolling around in the deepening winter.
A - space travel would be harder. The difficulty with travelling into the solar system is gravity, not the vacuum. You would still need to launch something with enough energy to escape the earths gravity. Aeroplanes are not escaping earth's gravity - they're constantly using fuel to stay a certain distance from the earth's surface but they do not have enough energy/power to reach escape velocity.
So if you filled the solar system with gas you wouldn't fix the gravity problem. What you would do is add more friction which would cause drag on space ships, and slow travel between destinations as well as require even more fuel than present. Once a ship is in space currently, "aerodynamics" is not an issue; it's all about gravity and velocity. Throw in air, and you have new problems in drag, shape and as a result likely fuel consumption to stay on course or reach as far as you want to.
Not exactly. All known means of propulsion are based off of throwing something backwards to propel yourself forwards. In a medium like air, you can grab the air in front of you and thrust it behind you. In space, you don’t have that option, you have to carry all of the propellant with you from the start and can’t get more unless someone brings it with you.
The air resistance would probably tear the Earth apart. Keep in mind we're whizzing around the sun at like 3000 MPH. All the planets move at different speeds, so something is getting wrecked by air resistance