As the saying goes - Problem is probably related to whatever was fucked with last.
Dealer might have messed up one of the connectors, pinched a wire or (more likely) popped a fuse while changing the battery. Will your cruise control engage? If it does, then it isn't a sticky brake or switch. Might be useful to know if both the taillights and brake lights are powered or not - can you see if both filaments are lit (dual filament 1157 bulb, I think)?
Be sure to post up once you find out the issue.
Note: To properly "fix" a sticky rear brake pedal, you have to physically remove it, clean the pivot and re-grease. Careful with the springs. Squirting a little lube in there will get you home but won't last. (Just sayin' - you may be doing it properly but I have seen people use a squirt of WD-40 and call it good.) Annual maintenance for me.