Changing the battery probably would increase the recording time, but I wouldn't recommend it - it is probably a non-standard shape, so you'll find it difficult to find one that fits.
It will take any type of MicroSD card. They are available in different classes (write speeds) and capacities. The speed of the card won't be much of an issue with this camera, as the resolution is quite low - you'll be able to get away with a cheap class 2 or class 4 card. It is worth going for an established brand imo (Sandisk or similar), as they have a significantly longer lifespan.