Your 2010, 13-inch, MacBook Pro will actually support up to 16GB max.
I think there is no way to predict how accurately the battery will be reported by a non-Apple operating system.
If the battery is original, then 12 or 13 years is a long time for a battery, particularly if it was stored for an extended period without occasional charging over that time.
With that in mind, then I suggest that if you are upgrading memory and boot drive, that you simply replace the battery at the same time.