Sure, it's fine to use the same stain for both sides. The important part is to use a different topcoat. Interior poly or varnsh for inside and exterior varnish for outside. You should do at least 3 coats of finish on the exterior. I like to use a sanding sealer on the inside as my first coat too....makes it easier to sand and gets the wood to stay smooth in fewer coats. Sure, pine is blotchy. There are wood conditioners available that are supposed to make the pine less blotchy. I have found that this depends on your eyes. I can't tell the difference between pine that was conditioned first and pine that was not. Maybe you can.

Thanks paintguy. That's really helpful.