Red eye is a result of having the flash close to the lens. Most small digital cameras will give you red eye. Some cameras have a red eye reduction that gives some flashes before the main flash to close the pupil so you don't get much red eye. I find that everyone ends up squinting or has their eyes closed when the main flash goes off so I don't use it, but it might work OK for your toddler. There is software with automatic red eye reduction that works well. I prefer to just do it manually if I am going to make a large print.
Pros don't get red eye because they use external flash units that are far away from the lens. If your camera has a hot shoe you can get the same results with an external flash. Bounce flash is especially nice indoors.
Available light photography indoors is almost impossible with most consumer digital cameras. Stabilization helps a lot, but it doesn't correct for subject movement. A tripod works, but it doesn't help for subject movement either. You are pretty much stuck with the flash for stills and need good light for movies. There aren't really any tricks that can overcome the basic limitations of the camera.