An underwater habitat or submersible can be designed to either operate at 1 atm or at the ambient pressure at depth.
1) Wouldn't the internal air pressure become equal to the water
pressure at the bottom of the sub where he enters?
If Ed Harris was able to enter through an open port in the bottom of the vessel then it must be the latter. But that is more usual for a habitat (that doesn't change depth very quickly) that for a moving submersible. I don't recall, but perhaps there was an airlock in the sub that allowed equalization before the inner hatch was opened.
2) If the above is true, than the internal air pressure of the sub
would exceed the external water pressure everywhere, except the floor,
inside the sub.
While there are significant changes in pressure in a column of water, there is not so in a gas column - because water is allot more dense than air. Consider over 20,000 ft accounts for about 14.7 psi at the surface of the earth, but the same pressure can be realized in a column of seawater only 33 feet! So in the sub, the floor, the ceiling almost all at the same pressure, the same pressure at the surface of the moon pool. That's why water doesn't enter and why the air doesn't force its way out. They are at a static equilibrium with one another.
3) Due to this positive pressure differential, any small cracks or
holes caused by damage to the sub should force air out, not let water
Not quite true. If the vessel is not intact, gas will indeed leak out (from any point above the surface of the moon pool) and the water from the moon pool will eventually replace it. Think about the grade school science demo of inverting a glass beneath the water, Try it with a clear plastic cup and poke a hole in the top of the cup. Gas will bubble out of the hole and the water will fill from below.