The distance between voxels centers in the row, column and slice directions is stored as pixdim[1], pixdim[2], pixdim[3] in the NIfTI header. Just make sure the image specifies NIFTI_UNITS_MM
  https://nifti.nimh.nih.gov/pub/dist/src/niftilib/nifti1.h
I am surprised that you list negative values, as usually these are absolute values. Not sure which tool described these. You can also determine the voxel size from the s-form or q-form, where negative values would create a negative determinant (e.g. an image that is mirrored with respect to canonical space).

Your formula looks fine, though I would use absolute values for volume. 

If you have a lot of images, you would be better off writing a Matlab or Python script. MRIcroGL can save images as either .nii, .nii.gz or .voi format - in reality the .voi format is just a .nii.gz (with the extension telling the software to treat it as a discrete rather than continuous image). A script like this one 
https://github.com/rordenlab/spmScripts/...
could easily be modified to report volumes for hundreds of images with whatever precision you prefer.