Super-late to replace the battery. Take the back cover off, pop the old battery out, pop the new one in. Remember when cell phones commonly had user-replacable batteries? Basically, that.
I cannot answer your question regarding the SD card, as I've had the same one in mine since launch. :/
I can say, my original 3DS got to where it was dead all the time. I assumed it was the battery just aging. Then, one day, I was messing around with it and popped the cover (a plastic, Marion themed thing) off of it - the back of the unit was bulging out where the battery was all puffy and ready to explodey. Called Nintendo to buy a new battery, they had me mail everything in, checked it all out to make sure there wasn't something wrong with the system, then sent it back with a new battery installed.