Certainly. First, if you look at Wikipedia, the batteries have different watt-hour ratings, which I assume would cause an issue. http://en.wikipedia.org/wiki/MacBook_Pro... I'm also pretty sure I remember reading an article about how the new batteries are slightly bigger (in physical size). And again with the full Logic Board switch, I'm thinking the possibility is very unlikely as the size / screw locations, etc might not match up. I don't have enough information to be sure though.
I think this is just how the iPhone OS was designed. Obviously when running off the battery, your phone will go into standby to save power. However, then the phone is plugged in, there really isn't a need to save that power. I've noticed this with my iPhone as well. If you want to put your phone in standby mode when charging, just hit the top button.
I don't think it should be draining that quickly. And you shouldn't have to drain the battery completely before charging it. It's a good idea to let it drain completely every couple of weeks however. Try a fresh start and restore the iPhone OS (and don't restore from a backup). I've found that my iPhone's battery lasts longer after a clean install. The OS might seem more responsive as well. I'd wait and see how the battery performs before you begin to add all your applications back, importing your contacts, etc.