It's not a space elevator! They're just very efficient rockets because nuclear explosions have high thrust. Chemical boosters launch the rocket up a bit, and then small nuclear bombs are released to push the rocket. Here is a short CGI demonstration.

Russia has indeed put mirrors in low Earth orbit. Znamya was a 20 meter diameter satellite that created a 5 kilometer diameter bright spot on the Earth. You could use this technology to get rid of long Arctic nights I guess but you'd have to place the satellite into a higher orbit so it's geostationary. Likewise, a solar shade could be made. If placed in the L1 point between the Earth and Sun, it would allow us to dim the sun slightly. I think it's a better idea than aerosol injection.

However, neither orbital mirrors nor solar shades are useful for power generation! The idea behind space-based solar is you would send a couple dozen large power plants into geosynchronous orbit and beam power down to Earth. Without an atmosphere, you receive more sunlight. And by being in GEO, the satellites only go dark for a short time once a year. The power would probably be generated with photovoltaics and beamed down using microwaves. A large rectenna on Earth would receive the power, and an exclusion zone would be cordoned off. Still, the extra heat would be pretty minor. Interestingly, the microwave power transmission uses the same frequency as Wi-Fi. Lasers do not penetrate the atmosphere well, although they probably make sense for beaming power from Mercury's orbit to the rest of the solar system.

There are many reasons why space based solar power does not currently exist and probably won't for some time. It requires a superheavy launch vehicle capable of regular missions and competence in manufacturing in space. It would probably be a trillion dollar investment, and America was more interested in building 1000 nuclear reactors instead (which it also didn't do. lmao). Cost is clearly an issue. These solar power plants make more sense when solar panels are extremely efficient or payload costs are very low. Getting payload costs low is taking a very long time unfortunately, and once solar panels become very efficient you may as well put them on Earth. That's what we are currently doing. Also these plants only last like 20-30 years for some reason before they have to be replaced. Pretty disappointing!

If you wanted to rain death upon your enemies from orbit, you should instead look at the this militarized proposal for Project Orion. A few of these ships could be put in orbit around Earth and each carry a gigaton of nuclear missiles and a few hundred nuclear shaped charges. The Kennedy admin found this was overkill tho.