I'm wondering which camera would be better to buy?

Nikon D5100 DX-Format Digital SLR Camera or the Sony alpha SLT-A37K DSLR Camera?

I'm looking for/wanting it for:
Fast Shutter Speed
Good Image Quality
Background Disfocusing
Family Photos
Indoors/Low Light
Landscape/Scenery
Sports/Action
Travel
Video
Weddings/Events
Wildlife photos

The only difference is that the Sony's better for video.

Find a local dealer and see which one feels better in your hands.

1) Even compact cameras can shoot 1/2000 of a second exposures, which is fast enough for action shots.
2) All DSLRs great image quality.
3) *defocusing* Depth-of-field depends on the lens.

Any DSLR can do wildlife photos and landscape things. But if you'll mainly be recording videos, buy an actual camcorders.