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Electric drones are well suited to 60,000 ft. The motors don't depend on oxygen nor the insulation capability of air. Some engineering challenges are cooling and battery capacity, but surface to 60,000 ft is certainly possible. I'd attempt with a two stage drone. The first would be optimized to fly from 0 to 35,000' with a much smaller upper stage optimized for high altitude flight. Both stages would run batteries in sequence and jettison the depleted packs in sequence. The intercept should take five to ten minutes. A single stage could probably do it too by jettisoning batteries but I'd expect the low altitude climb to suffer, and slower climbs, as in rocketry, use more energy.

No. The speed ratio for the same aerodynamic forces between sea level and 65,000 ft is 3.66 (square root of the density ratio). This translates directly to RPM. A high altitude drone would have oversized slow turning propellers at sea level. One airframe with fixed props could do it from an aerodynamic standpoint but there are motor efficiency considerations. A vertical climber also has a gravity tax similar to rockets so faster climbs are better. The gossamer aircraft required for high altitude flight would have excess drag during low altitude high speed flight, and a high speed climb is desirable for the aforementioned reason. The two stage approach with a much smaller upper stage alleviates the aerodynamic and motor matching problems, which is why I suggested it. The second stage also has a weight advantage since it's totally free of the first stage and starts essentially fresh at 35,000 ft.