Yes, if you have a source of lift, and you're within the atmosphere, then you can circumnavigate the globe without reaching orbital velocities. A ballistic rocket generally has no significant source of lift and spends nearly all of its time in space.
> Trajectories are shaped like parabolas, not ellipses.
All orbits are conic sections: either an ellipse, hyperbola, or a parabola (the limiting case between the two). But a parabolic or hyperbolic trajectory has escape velocity, whereas only an elliptic trajectory is planet-bound. The apogee of the ellipse is locally approximated well by a parabola, which is why for non-orbital mechanics ballistic trajectories are often modeled by a parabola, but all suborbital trajectories are actually ellipses, not parabolas. If the Earth were flat and the gravity vector were constant, then they would be actual parabolas.
Orbital mechanics is very counterintuitive. I recommend Fundamentals of Astrodynamics if you'd like to learn more, or play KSP rather than FSP.
> Well it's more like a quarter
On this you are correct, the map I looked at deceived me. :)
Math:
Reaching the edge of space requires 100 km altitude, or 1,000,000 m^2/s^2 of specific energy, which is 1,414 m/s velocity. Redirect that to a 45° angle and you have 1,000 m/s in both the vertical and horizontal direction, which comes close to maximizing your distance. That gives you 200 seconds of flight, which puts you 200 km downrange.
My experiments with Flappy Space Program contradict this. b^) It is perfectly possible to miss orbit even after a complete circumnavigation. Trajectories are shaped like parabolas, not ellipses.
New York to Tokyo is nearly half that circumference, so the energy required is nearly that of orbit.
Well it's more like a quarter, but even if it were half it would take significantly less energy than orbit.