Alien Bee's do shut off and then come back on when the strobe is fully reccycled
You can set them to come back on immediately, or wait for the capacitor to charge up.
*Is it okay to turn the modeling light off and just use the strobe once I know where the light will fall on my subject so that I don't have to worry about any loss of power or color change from the modeling light being on while the strobe fires.
Perfectly OK, but in all likelihood, it won't be a problem to leave them on.
Side Question: So I should not use a modeling light to light the backdrop. What should be used? A strobe with an umbrella attached or a strobe with a softbox attached or a strobe with a brawley box attached? If an unbrella what color; white, black or gold? I should mention this setup would be for a newborn session (cousin).
Again, you don't use a modeling light to light anything. If you are using a strobe to light your subject, you should be using a strobe to light the backdrop as well. (there are exceptions, but lets keep things simple).
Softboxes, umbrellas etc, are for softening the light. You typically want soft light on your subject, but you shouldn't need soft light on your backdrop...as long as you can get even coverage of light. Although, if you want to create a gradient or a fall off on the backdrop, then you might use something like that. It really depends what you want to acomplish.
When I'm going for a pure white backdrop, I use two lights and cross light the backdrop. This helps to give a nice even light and helps to prevent creases/wrinkles from throwing shadows. If I want to light just a part of the backdrop, I might use barn doors or a snoot or a grid, to keep the light just where I want it.
If I want a dark or black backdrop, I may not use a background light at all, and I may go out of my way to keep the other lights from hitting the background as well.