What's the problem with the battery/shutter? Nikon D3100?

(1) My Nikon D3100, on the first day of having it, after a few photos, just froze. After I pulled out the battery it started working again. I thought it doesn't matter, but once in a while it still does that and it's annoying. I think it's something with the battery, but how should I fix it?
(2) Sometimes when I point, after the auto-focus noise, when it should let me shoot, it doesn't, and I have to point a few more times, maybe even change the focus, for it to work. What is there to do?

Thank you in advance. It's in warranty, I could send it if it's serious, but I don't want to stay a lot of time without it.

Added (1). (2) It's not a light problem as far as I know, because when this occurs, the auto-focus is already fixed, it isn't searching for it any more.

Are you doing this in very low light?
If the light level is too low, the shutter speed will be very slow, as in many seconds.

If it's too dark and you have the focus illumination light disabled, the camera won't be able to achieve focus which means it won't let you take a picture.

If you go outside in day light, does your camera lock up?

There's a chance something is wrong with your camera, but not knowing your shooting situation makes this a difficult question to answer.

First day alone should have been enough reason to return the camera back to the store and demand for a replacement. I believe you won't be able to do that after just a few days (a week or less) after purchase. After the replacement period, the camera will be sent back to the factory for repair if under warranty.

Anyways, check for what Qrk said. Shooting in low light does have the same effect. Shooting at subjects in very close distances also does the same.

To test, set camera to A mode, AF on, EV to 0, white balance to Auto, and ISO to Auto. Go out and look for a brightly colored car parked under the sun then shoot from about 10 feet. If that doesn't go on normally (fast) then something is definitely wrong with your camera.