Answer to: Where to get good batteries?Ifixit is the right place
Answer to: MacBook Pro (Mid 2012 - Model: A1278) Random Shutdown on BATTERY ONLY!I would recommend you to remove all the unsupported applications (if you are running any) and reinstall MacOS, might help!
Answer to: Would Replacing processor to Core i7 make my computer faster?You cannot replace your processor as easily as a Hard disk or Ram. I would recommend you to replace your hard drive from a preinstalled one to a Solid State Hard Drive (SSD) and Ram to 8 or 16 GB ~ which ever your MBP supports. Note : SSDs can vary in price range, i would recommend you Samsung 850 EVO 500GB 2.5-Inch SATA III Internal SSD (MZ-75E500B/AM) available on Amazon for $129.99