Need(ed) More Input (Second edit includes many more details)
Specific orbital behaviour would depend on a lot of factors, but some of your questions can be answered regardless.
You've not indicated whether the new moon is smaller than the existing one, but for the sake of argument we'll assume smaller because something as massive as the moon between the current moon and the Earth would definitely cause some interesting orbital perturbations that are potentially beyond the scope of the question. That said, even a small moon is going to be visible in the sky, probably during day and night, as with the current moon, while it's illuminated by the sun. (The ISS is visible from Earth and it's much smaller than the moon.)
Something the mass of the moon or smaller, orbiting inside the current moon's orbit, will definitely be tidally locked unless this moon is a recent acquisition. If it's been orbiting with its larger cousin this whole time, it will be tidally locked.
The place where this gets interesting is that our moon was formed by the collision of a third body with Earth, flinging the molten moon into space. We didn't have a substantial formation disc, as the gas giants did for their moons, so there's no reason to assume that two moons are necessarily coplanar. If they're not, the answer to "when would they align" might be "almost never". If they are, however, the answer is just "at common multiples of their orbit, on different points on the Earth's equator".
So to actually calculate when they would align, we'd need to know what InnerMoon's mass is, its orbital radius, and whether it's coplanar with OuterMoon.
Edit: It's also worth noting that the Moon's orbit started out ~20 000 km from Earth, so it's hard to imagine where this second moon might have slotted in without being torn apart by tidal forces, unless it arrived/was formed later.
Edit the second, in light of comments:
If the inner moon had half the radius of the outer moon, and we naively assume that the densities of the two moons are homogeneous and identical, the inner moon would be an eighth the mass of the outer moon.
If the inner moon's orbit is half that of the outer moon, e can then ignore moon-moon interactions (we shouldn't, but it's easier in the short term if we do), to calculate the orbital period:
$$ T = 2\pi \sqrt{\frac{\alpha^3}{GM} } $$
the only significant changing value is alpha, the semi-major axis, so the inner moon's orbital period would be $\sqrt{\frac{1}{8}}$ that of the outer moon. So since the moon's orbit is once every 27.3 days, the inner moon would whip 'round every ~9.65 days. Assuming that they're coplanar and not retrograde to each other, they would appear aligned over some spot on the equator every 14.93 days.
Angular radius, when viewed from earth, is a straight ratio of orbital radius and orbiting body radius, so since the inner moon is half the size but twice as close, it would appear the same size as the outer moon.
Now, having an extra nine quintillion tonnes in orbit would not make things quite this straightforward, but we're assuming that the orbits, at the moment of your story, are in this state.
