Why does a cathode have to be heated to emit electrons? Considering that electrons are highly mobile inside of a metal, why do they have such a tough time getting out at the edge of it and continuing their trip ballistically?
 A: Matter is held together by the electrical attraction between the electrons and the nuclei. Within the bulk of a solid or liquid, an electron feels these attractions from all directions equally, and therefore the force it experiences equals zero on the average. But if an electron finds itself at the surface, this isotropy is broken. The electron feels attractive forces toward the interior, which are not canceled out by any forces from the outside. Normally this causes any electron that impinges on the surface to be reflected back in like a pool ball hitting a cushion. To extract the electron out beyond the surface, you have to do a certain amount of work, called the work function $W$. The work function for a metal is typically about 5 eV.

Why does a cathode have to be heated to emit electrons?

It's not actually true that it has to be heated -- cold-cathode devices do exist, and thermionic emission does occur at all temperatures. However, at room temperature, $kT\approx 0.03\ eV$, which is much smaller than $W$. That means that only a very tiny fraction of the electrons have more energy than $W$. The probability of having an energy $W$ at temperature $kT$ goes like $e^{-W/kT}$, and Richardson found in 1901 that the current from a cathode, in the absence of an externally applied electric field, was proportional to $T^2e^{-W/kT}$.
A: Your main mistake is that you think about electrons being moving along the lattice ballistically.
There at least two answers to this question: the classical one, and the QM one.
The classical answer is that even when the electrons are "highly mobile inside of a metal", they just hop between adjacent atoms. When it is in the atom, it resides on one of the allowed potential energy levels - exactly the same situation as encountered with a single atom. In order to remove the electron from the atom it resides in, you must supply the required amount of energy (which is called Work Function; Work Function of a material differs from ionization energy of material's element).
The QM answer is that when you stick all these atoms together, the orbitals of the valence electrons change such that they span the entire lattice (in fact, these orbitals even go slightly beyond the volume defined by the lattice). The above statement means that the probability of finding an electron far from the lattice it belongs to is negligible.
