How long have you had your 2010 MBPro?
New batteries, which will be third-party, sometimes fail
Your symptoms continue to make me think that you have one or both RAM sticks bad.
Bad RAM can cause some really strange issues
The memory slots in those older can turn out to be fragile.
MacBook Pro can be frustratingly picky about RAM.
Make sure it is actually a supported RAM type (PC3-8500 SO-DIMM)
The hard drive cable (that innocent-looking ribbon cable) definitely can cause problems with booting -- and it is a common issue on MacBookPros.
Best way to test RAM is testing in another similar Macbook Pro, where you can boot successfully, and test with something like MemTest.
Another alternative is to "try known-good RAM, of the correct type, from a known good manufacturer, such as Crucial"
Still no booting after that? Logic board then.
New batteries, which will be third-party, sometimes fail
Your symptoms continue to make me think that you have one or both RAM sticks bad.
Bad RAM can cause some really strange issues
The memory slots in those older can turn out to be fragile.
MacBook Pro can be frustratingly picky about RAM.
Make sure it is actually a supported RAM type (PC3-8500 SO-DIMM)
The hard drive cable (that innocent-looking ribbon cable) definitely can cause problems with booting -- and it is a common issue on MacBookPros.
Best way to test RAM is testing in another similar Macbook Pro, where you can boot successfully, and test with something like MemTest.
Another alternative is to "try known-good RAM, of the correct type, from a known good manufacturer, such as Crucial"
Still no booting after that? Logic board then.