Early "F" engine coils were grounded to run. Later "F" engine coils were grounded to stop. The earlier coils are no longer being made. Your replacement coil is most probably the grounded to stop variety and it can still be used on the earlier "F" engine mowers, just turn the on-off switch to "OFF" to run, and "ON" to stop. I know it is confusing. But try the coil with the kill switch set to the off position. I bet you get a spark.
There is no battery. It is a self-contained magneto-type ignition module with no points, on an "F" engine. Excited by the magnets in the flywheel. Even it you have a model with electric start...the whole battery-charger-electric-start system is totally separate from the ignition coil/magneto. There is a separate coil, totally independent mounted near the flywheel that generates a charging current from the magnets in the flywheel. Not connected to the running of the engine at all, and the engine can run just fine with all the electric start stuff removed.