Is it possible to launch a projectile directly from the surface of a spherical, airless planet into a closed orbit which does not touch the surface of that planet? If so, what combination of launch angle and launch velocity would be required? Assume that the projectile has negligible mass and the planet is the only object with a significant gravitational effect on the projectile.
Classical Newtonian orbits are closed: they will come back to where they started.
If the planet was there (and in your problem statement that seems to be an assumption), your orbiting object is coming back to the planet.
I've always rather liked Newton's cannon on a mountain idea, because (assuming we could neglect air resistance) you'd at least have 80 odd minutes to move the cannon.
In absolute terms no, but practically yes. That is a fixed delta-v is required to achieve a minimal orbit, but if that delta-v is nearly instantaineously applied at the launch site, then the launch site will remain on the orbit (which is bad, right?). Practically at the apopapsis (high point of the orbit) a second paritial or complete circularisation burn would be used, lifting the periapsis (low point) off the launch site.
You can calculate the periapsis orbital speed from this equation https://en.wikipedia.org/wiki/Orbital_speed#Precise_orbital_speed but you have to choose a semi-major axis to be some margin greater than the planetary diameter.