"Not require special batteries"
I take that to mean you want a camera that uses AA batteries.

That presents a problem - The cameras with the very fastest shutter lag do not use AA batteries.

But the Canon A1100 camera does use AA batteries and has a shutter lag of 0.4 second.
That is considered faster than average (which is 0.5 second).

I am assuming you are going to take these photos outdoors.
If you have plans to photo indoors with flash, you may be disappointed, because the flash will cause ghost eyes on dogs.
That is the dog version of red-eye we see in people photos.
The only way to prevent it is to have someone distract the dog so that it is not looking at the camera.

I don't know of any digital camera that can directly download to a digital picture frame.
You can download to a computer and then by using a card reader/writer on the computer, you can write photo files to a memory card that can be inserted into a picture frame.