If the Classic stops at Quizzical Floppy it is nowhere near the stage at which mouse-tracking has been enabled, so unresponsiveness of the arrow is not-at-all remarkable. You don't mention whether you have an SE with dual 800kB FDDs, a ditto with an added-in (third-party) HDD, or an FDHD with 'official' (factory-installed) HDD and FDD. Probably not the latter if the FDD is only 800kB, but how do you know that there is an HDD? Do you hear it?
It sounds as if you are at the same starting point as many another new father of an old Mac. Replace battery. Zap PRAM from cold start. Continue startup with extensions off. If you get to the desktop, look for the System Folder on the HDD. If no volume mounts, start up from a Disk Tools floppy and fix the HDD. If there is no HDD (which you will see when you replace the battery, anyway), you're the proud owner of a floppy-driven Mac. Welcome to 1990.