How to ensure the planes landing and take-off in the same runways safety?
No one knows.
Note this isn't exactly what is being proposed
By dividing it in two with a generous central buffer zone, we create two in-line runways, allowing aircraft to simultaneously land on one and take off on the other.
So that isn't simultaneous landing and taking off on the same runway.
Currently only one LHR runway is used for take-offs at any time. The runway is switched to provide respite from noise to local residents in the take-off path.
Also the surrounding airspace isn't set up in a way that would allow separation for concurrent take-off from two runways. A major reorganisation would be needed of airspace in that region of the UK.
I believe that currently, simultaneous take-off require parallel runways a mile or more apart.
How many flight can be handled by such design
No one knows. Maybe a c 46% increase from 480,000 to 700,000 air transport movements a year. See reference below. Bear in mind that Heathrow is subject to a night-time curfew.
How efficient is it compared to parallel runways?
No one knows.
According to the airports commission in 2014:
The concept of an in-line runway is unconventional, has never been used elsewhare and is not envisaged in international, EU or national regulatory instruments, standards or recommendations for safe design or operation.