Summary: No, this won't work in any way, shape or form. And if you did get it to work, it'd be more expensive than using rockets.
10 reasons it won't work:
Aircraft can get to an altitude of ~30 km. Above that, the atmosphere is so thin you need unfeasibly large wings to provide enough lift. An aircraft that flies next to a rail is no different in this regard.
Then there's the method for transferring power from the rail to the aircraft. This has been studied extensively: trains use the same method to transfer power from their overhead catenary wires. At a speed of 500 km/h this is already problematic, as the SNCF found when preparing for the world speed record with their TGV. The pantograph presses against the power wire/rail. This moves the power wire/rail. At low speed, this movement is small enough that the pantograph remains in constant contact with the wire.
At high speed, the pantograph can't follow the wire reliably anymore so you get intermittent contact. Applying more (mechanical) tension to the wire reduces how much the wire moves around, but this is limited by the mechanical strength of the wire. The highest speed ever achieved with such a power transfer method is ~580 km/h or about 200 m/s.
Then there's the distance between the aircraft and the rail. This has to be small enough to allow a structure like a pantograph to bridge the gap. But it has to be large enough to prevent the aircraft from hitting the rail when it encounters turbulence (which can easily displace the aircraft tens of meters).
Even if you can overcome the turbulence problem, you've carried your rocket to 20 km altitude and 200 m/s. You need 200 km and 8 km/s, so all that effort has given a very small reduction of the rocket size.
Airplane-launched rockets exist (Pegasus, Virgin LauncherOne) but they aren't common. Most rocket companies don't think it's worth bothering with an aircraft as the first stage. An electric-powered aircraft would add several more hurdles to an already difficult problem.
It seems you're trying to make the launch cheaper by saving fuel. But the fuel cost is a negligible part of the cost of a rocket. A SpaceX Falcon 9 costs \$60M to launch. Only \$200k of that is the fuel cost.
Even if you could accelerate to 1000 m/s on your launch system, that leaves 7000 m/s that must be provided by the booster. 7/8 of a Falcon 9 is still a \$60M rocket, it'll just be a bit shorter than the current F9 because it needs less fuel. The only cost advantage is in a barrel section for the fuel tanks you can omit, and those are cheap parts.
So in order to save a few hundred thousand per launch, you're proposing a rail system and dozens of blimps. Launch rates would have to be huge to make that economically feasible. Until then, using an aircraft as the first stage is a better option, and even that has its limitations, as no aircraft exist that are large enough to launch e.g. a 4-ton communications satellite into GEO.
New: cost, wight, volume calculation
Catenary wire as used by railroads weighs on the order of 600 kg/km.
Bu60-1 was 60,000 m3, that's 50 m in diameter and had a payload of 4.6 kg. So a balloon capable of supporting 500 kg would need a volume of ~60 million m3, that's a diameter of 500 m.
The largest high-altitude balloon is 0.5 million m3 and has a lifetime of 30 days.
So your balloons would need to be 10 times larger than the largest high-altitude balloon ever made, and you'd need to replace them every 30 days.
The string needs to be at a shallow angle to prevent the aircraft from stalling. Say a 1/10 grade. To reach 50 km altitude, you need a string 500 km long.
- A catenary 500 km long doesn't work: railroad catenaries are supplied with power at regular intervals, at 500 km the voltage drop would be severe.
- to fill the balloons, you need 3 billion m3 of helium, that's 30 times the world annual production. Cost is on the order of billions of dollars.
- I've ignored wind effects: high altitude wind exerts lots of force on the balloons and the wire.