It is not a battery issue.
It is no memory issue - when your PC can modify the N900, the N900 should be careful with messing around with the same. Windows is not made for multi-user, while Linux is and will exclude others from messing around with the same things.

There is no need to flash it, tools are available to find errors in the file system that you may get if you stop the phone by just removing the battery instead of running a proper "Switch Off". Use e.g. "TestDisk" to verify and rebuild the file system.

If you cannot reach the phone from your PC, the problem is usually your PC and the Windows version you use. Take 2GB memory stick and place a bootable version of Ubuntu or Mint on this, and install the Flasher and whatever you need (like hfs+ if you use Mac) and use the Linux drivers. Linux can access the file system on your computer and can back up "MyDocs" - also the "hidden" files.

You get a very specific error message if the N900 is to blame and the eMMC is really not available. You are not there.