"Where to get good batteries?"に関する回答Ifixit is the right place
"MacBook Pro (Mid 2012 - Model: A1278) Random Shutdown on BATTERY ONLY!"に関する回答I would recommend you to remove all the unsupported applications (if you are running any) and reinstall MacOS, might help!
"Would Replacing processor to Core i7 make my computer faster?"に関する回答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