These two images are telling:
http://pop.h-cdn.co/assets/15/52/1600x800/landscape-14508878... http://pop.h-cdn.co/assets/15/52/1450888612-bevsx.jpg
http://pop.h-cdn.co/assets/15/52/1600x800/landscape-14508878... http://pop.h-cdn.co/assets/15/52/1450888612-bevsx.jpg
Even in an ASDS landing, it exceeds the Kármán line. I think the typical apogee is around 120-130 km.
Stage sep happens below the line, however.
Fuel is heavy, and most of it is spent on... lifting fuel.
Having a nuclear explosion for example behind the rocket would surely be enough to push it into space... even on Jupiter.
Not quite, it's all about mass flow.
It's not about the amount of thrust. It's about being able to carry enough fuel to sustain that thrust long enough, without the weight of the fuel making the rocket too heavy to lift high enough to reach orbit.