Is the Nikon 35mm 1.8 good for indoor or outdoor sports?

It's a bit wide for portraits - a 50mm would have been better, and it's no good for sports unless you are right next to them.

I wouldn't use a 35mm lens for either of those things. It's certainly not the optimal focal length for either of those types of photo. I find it an odd choice for those purposes.

Shooting with the lens wide open will cause a shallow depth of field, that's what's causing your focus problems.

A 35mm lens on your cropped sensor camera works out at around 52mm (35mm equivalent) which is pretty close to what some people call a normal lens - neither telephoto, nor wide angle. For portraits a moderate telephoto fast prime is better. For sports a long telephoto zoom with a wide maximum aperture is best.

Depends how far you are from the action. It isn't the first lens that comes to mind. Normally you would go for short to mid telephoto indoors (70-300) and go longer outdoors.

But it depends on the sport.

No. Unless you're 5 feet away from them.