What happens if the electron in the lowest energy level of an atom loses its energy? Excited states spontaneously give of energy and transition to a lower energy state. Why can't an electron in the lowest energy level spontaneously lose energy and fall into the nucleus? Wouldn't that be an even lower level as it maximizes the coulombic attraction between electron and nucleus?
By definition of lowest energy level, you cannot go any lower!
As you already suggested, by classical mechanics one could imagine a situation where the electron is even closer to the nucleus (until it falls into it), than the lowest energy level (= quantum mechanical ground state) actually has. But quantum mechanics does not allow this. This is one of the key aspects of quantum mechanics.
This also relates to having no energy level between the ground state and the first excited state. Energy levels are discrete or quantized. Again in classical mechanics, where energies are continuous, we could describe some energy level in between. But quantum mechanics simply does not allow this.
The difference between what would classically be possible and what quantum mechanics allows for, is commonly also known as zero point energy. (And anyone who tries to sell you on getting some "free energy" out of this is wrong, because the whole point of the term zero point energy is, that you cannot go any lower than that.)
Why does quantum mechanics say there is a lowest energy level?
It does so, because it needs to represents reality. Remember that quantum mechanics is not the ultimate truth, but just a theory developed to give a working description of reality. To do so, it needs to assume a couple of things.
Let's assume for a moment there would be no lower limit and an electron could go arbitrarily close to a nucleus, as classical mechanics allows it to do. This would make atoms unstable. The electron would constantly emit light (as the Maxwells equations for a moving charge require) and spiral into the nucleus. However this conflicts with reality, as it would mean: no stable atoms, no stable molecule, no chemistry, no biology, no humans, nobody to ask that question in the first place. Additionally, it could not explain why emission spectra look the way they do.
In order to reflect these experimental facts, quantum mechanics can only be a viable theory if it asserts energy levels are discrete and there is a lowest possible energy level.
Whether there is some deeper explanation where this discretized nature is coming from is unkown. A theory going beyond quantum mechanics might explain it, but this is an area of ongoing research in the physics community. But chemistry is not really concerned with this, as quantum mechanics is all it needs.