What frequency level do older hearing aid batteries work? Is it normal to hear the whine from someone else's hearing aid battery?

"whine" from a hearing aid might be feedback. This is when the hearing aid's mic hears the audio output of the same hearing aid and similar to when someone points a mic at a PA system speaker and gets that squeal. Hearing aid batteries do not make noise.

Batteries have nothing whatsoever to do with frequency.

Period.

Batteries produce pure DC, which in AC terms is "zero Hertz".

On an oscilloscope, it would appear as a flat line.

If there is a difference in performance that depends on the cell being used,

it is due ONLY to the exact voltage and current available from specific cells.

If there is a whine, it is caused by something other than the cell(s).

Batteries do not whine in any way, shape or form. If you are picking up a whine on your hearing aid it could be a fault, or it could be radio interference from somewhere. Hard to tell without more details.