Edit: A bounty was offered; Later, I answered my question - see below the looong math answer. I accepted it, because it's the only one to calculate the outcome with all the parameters included. I'll award the bounty manually to the answer that best addresses the question (except mine).
This is the second, updated version of The Project Sahara my grandfather made with me:
The European countries in cooperation with USA, Japan, and other economically strong states will give money and brains to build and manage this project. On the western African coast, close to the equator, where the rocky Sahara desert is, giant fields of solar panels and thermosiphons (water heaters/boilers) will be built.
Nowadays solar panels have an efficiency of about 5%, that is a power of cca 70W/m^2, and water heaters have an efficiency of cca 30%, that is 400W/m^2 from the Sun (not in electricity this time). If we had 1000 square km with 50% water heaters, we would get a power of 235GW - that is 1020TJ (terajoules; 1 terajoul is 1000000000000 joules) of energy every day (12 hours). Without using any otherwise usable land.
The energy would be used to do this:
boil seawater using the water heaters to get (distilled) water and salt
sell the salt (salt prices fall down; From about $0.2/kg to about 0.05/kg, then the other salt-selling companies fall, and the salt prices go up again)
export some of the water across Africa
export some of the electricity across Africa
using the electricity it will break down water into oxygen and hydrogen, and using CO2 from the air and the hydrogen, produce methane (CH4)
export methane as a fuel for cars, etc., instead of petrol - it will be exported using pipes, as petrol is - petrol prices fall, and the states and companies selling it will suffer a crisis; ISIS goes short of money.
sell some of the oxygen to whoever wants it, and release the rest into the atmosphere - it's an unwanted product.
This project has many advantages:
methane is easy to store, especially long-term; Hydrogen isn't, and electricity isn't at all
logistics: it will be transported as easily as petrol, maybe easier
burning methane... well, burning methane will produce again the CO2 used to make it - no change overall
a lot (thousands, actually) of working spaces created. This could literally employ a whole slum city to maintain the panels/heaters - sweeping panels and guarding entrances doesn't need qualified workers. This would update the economical and political situation in whole Africa
new technologies. New technologies would be invented along this project - like it was during NASA's moon-conquering program
no more Russia and Saudi Arabia dictating fuel prices
no need to push away the locals - there are no locals, instead, people will be encouraged to come
Is such a giant project achievable? I know it will never happen in reality, but that's because if the political situation, but I'm asking about the technical side of this.
Is this feasible?
And importantly, how can I improve this project?
Also, thanks to @TrEs-2b for his interest and support of this project: What are the political outcomes of the Sahara Project?