This question has several facets to it, I'm going to try to answer as many as I can, edits to follow.
Lifting Gas Availability
Helium is a limited resource, although a lot of it is currently produced as a side effect of natural gas extraction. 2008 production was ~169 million standard cubic meters (SCM) of helium. This would not even be close to enough for your mega-structure. 100km x 100km x 100m = 1x10^12 cubic meters or 6000 times the yearly production. (I assumed 100m thickness it could easily be bigger depending on the density of your craft)
Hydrogen however is not very limiting at all, as it is abundantly available in water, requiring only electricity to extract it. So despite the flammability issues and diffusion problem, you are going to have to use hydrogen. On the plus side it is the most effective lifting gas available.
You could also use hot air, which simply requires heat.
So I'm going to do a bunch of big approximations to get a back of the envelope calculation for this using a quick and dirty energy balance model.
So the Sun provides ~1000 W per square meter, but only when it's shining and there is a lot of variance over day, night, seasonally, and with latitude. Average all of that variability out over a year and most places on Earth's surface will see an average of around 250 W/m^2. (Using this lower number will underestimate cooling rate during the day)
Some relevant properties of Air:
Specific Heat: ~1.0 kJ/kg K
(varies a lot with altitude and temperature but a rough average for a quick calculation)
So if we remove 250 W/m^2 from a column of air 15 km tall we can figure a rough temperature drop.
Mass of the column of air = 15km * 10km * 10km * 1kg/m^3 = 1.5 x 10^12 kg of air
Rate of energy removal = 250 W/m^2 * 10km * 10km = 2.5 x 10^10 W
E = mcT to power P = mcT/s
solve for T/s = P/(m*c) = .0000167°K/s or .001°C/minute or 1.44°C/day
To get the final coldest temperature you would need to factor in convective heat transfer involving the mixing with warm from air outside of the shade zone (this cooling would cause winds to develop. Anti-cyclone!) as well as radiative heat transfer and heat from geothermal sources, and water motion for oceans or large bodies of water, in general it would be a really complex set of calculations and very specific to the location and timing of the shade placement.
But to simplify it you are essentially creating a stable stationary cold front, which can have temperature drops of up to 30 °C (54 °F) so this would be a likely upper bound for temperature drop from ambient levels in the region under the shade.
Balloon tethers are definitely possible and in common usage at lower altitudes.
Barrage Balloons were used to raise nets of metal cables into the air to impede enemy aircraft. They were raised to ~4,500 m (15,000 ft.), modern tethered balloons are used primarily for military surveillance and reconnaissance, as well as communications. Some current military systems operate at altitudes of 15,000 ft. with 25,000 ft. long tethers.
Longer tethers and higher altitudes are likely possible with existing materials and even better for possible near future materials. For tether material strength over those distances the important factor is known as Specific strength or breaking length, how long a tether can be to support its own weight. Modern existing materials can be quite long Kevlar and Carbon fiber have a breaking length of ~250 km (for reference 50,000 ft is ~15 km).
The bigger issue for tether strength would be if it was used to resist the force of the wind on the balloon structure, but if it is a powered structure capable of some maneuverability, or the tether is not secured to the ground these forces could be mostly negated.
Most of these issues would be an engineering design problem, but not impossible, blimps and zeppelins have been shown to function so it definitely can be done, it just needs to be scaled up a lot.
It won't be fast, will need to be aerodynamically streamlined to resist winds, and will likely not be very maneuverable. Your best bet on travel would be to find favorable winds, by changing altitude, and going with the wind.
As for solar power, the Hindenburg used 4x Daimler-Benz DB 602 (LOF-6) diesel engines each 890 kW (1,200 hp), so total power of 3500 kW.
The Hindenburg had a diameter of 41m x 245m long; so using only the top half of the surface for solar cells (π * D * L)/2 provides roughly 16000 m of surface. Using solar cells (20% efficient) to provide 200 W/m then provides 3200kW.
So comparable instantaneous power, but it would likely be underpowered if requiring continuous thrust for any length of time. And it would be worse when considering storage losses required for night running. As for energy storage hydrogen fuel cells would seem like the obvious choice over batteries, given the likely use of a hydrogen lift gas.