Let's say you want to use helium as a lifting gas. Hydrogen can be used too, the numbers don't really change. Helium has a molecular mass about 14% the molecular mass of air at the same conditions (pressure and temperature).
Imagine now you build an airship the size of the Hindenburg. About 200000 m3. Each m3 provides a lift of very nearly 1kg, so you can lift with such an airship 200 tons of stuff, but that includes the balloon itself, the cables, everything.
Now imagine the balloon has folds, like an accordion. As it goes up, you expand the balloon until it becomes some sort of very long and very fat sausage. It could even be mile-long.
Where will the balloon stop ascending. Let's say you design the internal pressure to exceed the external pressure by about 1% at sea level. The balloon stops roughly where this 0.01 atm pressure is about 7 times higher than the external pressure (because helium compressed 7 times more than air has the same mass). That happens where the atmospheric pressure is about 0.0015 its sea level value, and this happens at about 50 km height [1]. You could say you are half way to space.
Once you get to this point, you need another source of lift. This is airplane type of lift, except that you don't need wings, because the entire balloon can be tilted slightly upwards and function like a gigantic wing. In order to not suffer from the tyranny of the rocket equation, you need reaction mass. For this you have an outer shell around the balloon that works as a funnel. You scoop whatever air is at that altitude and direct it to inside of a combustion chamber, and you push it at high velocity on the other side. You accelerate slowly over many days. Maybe during the day you use solar panels to heat up the air, and during nights you use hydrogen that you have onboard, and which has a huge energy density. On second thought you might even use hydrogen as a lifting gas, so you have plenty of it at hand.
As you accelerate at some point you reach supersonic and then hypersonic speeds. We are used to thinking these speeds are very destructive to the airframe, but at extremely high altitudes, where the air density is, let's say 1/10000 the density at sea level, maybe they are not so destructive. As you pick up speed, some of the lift comes from the orbital motion. Little by little you keep going up and keep speeding up. I don't see why you can't reach 7 km/s.
Good question. I googled and found out that the highest clouds are the noctilucent clouds. They don’t contain dust, but tiny ice crystals, which are less than 0.1 microns in size. They can be found as high as 80km. Maybe you calibrate the airship speed so that hitting such a cloud in the “danger zone” does not damage the balloon. When you are above 80 km, you step on the gas.
I don't know the answer to that. But there's a pretty good chance the dust would get deflected in the boundary layer. The more problematic part is when you got to the orbit and there is no more air to form this boundary layer, and specks of dust will hit the balloon directly. I suspect that the microscopic holes that they form will let some of the gas out, but not enough to make a difference.
