Based on your requirements a DSLR is necessary.

The main point is your focus on indoor photography, this requires at least a large sensor camera, so either a DSLR or SLD. No point-and-shoot gets close to those in terms of performance, although you may have seen very expensive fixed-lens cameras like the Fuji X100 which can do it, but that one has NO zoom.

The second point is speed. There three parts to this:

1. Shutter-lag which is the time the camera takes to respond to you pressing the button. DSLRs lead here by a wide margin and this is probably most important when shooting kids because it lets you capture a precise moment.
2. Autofocus speed. This is the time it takes for the camera to focus on your subject. Again DSLRs focus faster than all point-and-shoots.
3. Continuous shooting. This is the speed at which images are shot continuously and where CMOS-based point-and-shoots lead by a huge margin. The top DSLR shoots at 11 FPS while the top P&S can do 60 FPS.

Understandably you are concerned by the lack of zoom. Zoom is not part of a DSLR but a property of its lens. The good thing is that it can be bought incrementally. Now, you can get a DSLR with a prime lens (no zoom at all) which is best for low-light and speed and buy zooms as you go along and can afford it.

Modern DSLRs have awesome video quality. Even though some can focus automatically while filming, doing so is poor (slow, noisy and disturbing). You really should learn to focus manually while filming. The shallow depth-of-field gives a cinematic feel but that also means that focus needs far more attention.

The only intermediate solution is to get an Sony SLT camera, like the SLT-A55, that takes interchangeable lenses, uses the same sensor as a DSLR but focuses much faster while filming. It also shoots continuously at 10 FPS with restrictions (or 7 FPS without).