On a laptop battery failure is a common cause of anything.
How did you test the battery?
Did you try direct substitution?
I never personally replaced a Hard Drive in a Mac Book
Here is what is found on the Apple site.
https://discussions.apple.com/thread/5171107?tstart=0If you have a 2009 model then you must of gotten the original system DVD discs that shipped with every Mac made before Lion came out. If you no longer have them you can order a replacement set from Apple for a small charge. There is no other way to reinstall OS X on your model Mac without them. Then you would need to upgrade to Snow Leopard !0.6.6 or above to get access to the Mac App store to reinstall Mountain Lion. That is unless you have a Time Machine backup made when Mountain Lion was installed. If you do have a TM backup you can boot the system from the Recovery HD system on the TM backup drive and then restore that TM backup to your new HDD.
So, if that is true, you can not boot it from a USB, You must have the DVD.
That is the best I can do for you.