Re the need to have jet engines in order to fly at the speed of sound.
Whilst this is true of current technology, it is not necessarily true. What is required is enough thrust to accelerate the vehicle to the required velocity. In general this means moving a lot of air with enough energy.
Propellers are excellent at moving lots of air, but lose out on efficiency at high altitude (due to air density) and at high speed, propeller tips travelling at or exceeding the speed of sound just don't work very well.
Jet engines on the other hand work quite well at altitude, they too have problems at supersonic speeds. The problem is the airflow through a jet engine at supersonic speeds can cause flame out and loss of power. I did learn about this millions of years ago during my apprenticeship, so I am a little rusty. As far as I am aware, must supersonic capable jet aircraft employ ramps in the air intakes to slow the airflow into the engine to sub-sonic speeds. Concorde did this and it was all controlled using analogue computers!
So all you need (sic) is a means of moving subsonic air in very large volumes backwards - One device capable of this that needs scaling up is the ducted fan. A large proportion of Radio control model aircraft employ this method of propulsion.
Could it work for a VTOL aircraft - Its all about thrust to weight. Just move enough air and voila.
I suspect the main issue will be energy density vs mass. The ideal would probably be high energy for a low mass. Hmmmm...