Little known fun fact, but Earth actually has a second moon. It's significantly smaller than our central moon (about 164 feet across) and is essentially an asteroid. It corkscrews in a bizarre orbit around Earth, since the Earth's gravitational pull is playing tug-of-war with the sun's.
That to say...if you're wanting a second moon that doesn't massively affect the planet, you're going to need a pretty small moon, since the larger the mass, the more effect it has on tides, etc. But for it to be visible with the naked eye, it either needs to be insanely bright in the night sky, or be much, much closer to the planet than the "major moon." (I'd actually go with both bright and near).
We can see satellites in the sky already, so if you have a moon the size of a large satellite at about the same distance, you could potentially see it. To make it bright, it would have to be made of something reflective, or be capable of giving off its own energy. Currently I don't know of anything in the cosmos that can create such an energy at that size (quasars are abnormally bright for their size, but even they're massive compare to what you're looking for), and other small, bright objects like comets would be hard to find circling the earth (for a more in-depth reason for this, here's another Stack Exchange answer dealing specifically with comets).
Perhaps your second moon actually is a man-made satellite, but the planet's population doesn't realize—it could either have been space junk from another planet, or it was sent up by a previous civilization...all depends on the history of your world. But a satellite made of metal and reflective panels may be brighter than an asteroid or any other natural body orbiting.
For your secondary questions, their answers would depend on the size and how close your moon orbited to the earth, though I would think your small moon wouldn't go behind the large moon—instead it would pass in front.
For bonus reading: https://www.newscientist.com/lastword/mg24232301-200-has-all-the-hardware-sent-into-space-affected-earths-gravity/