This apt just sold for $24 million.

To judge by the floor plan, it's huge:
http://img.streeteasy.com/nyc/attachment/show/732717-834-5-avenue-new_york.jpg

But what is going on with all those extra random toilets?

The one at the far eastern end of the 3rd floor is an extra for servants, who otherwise would have only one bath for four bedrooms: http://www.columbia.edu/cgi-bin/dlo?obj=ldpd_YR_1350_MH_006_001&size=large

That wing on the 4th floor was extra maids' rooms for the whole building, so can't tell. Probably same story.

The one under the back stairs was added later and must be for catering staff, so they wouldn't need to use the guest lav or one of the maid's baths. For sink they have the mop sink across the hall.

BTW, the Columbia plans show the building when it was just a mid-block lot, so B there is C as built. The developer bought the south corner lot during construction, and added the A line. Layouts didn't change, though.

There seem to be three stray toilets there not in bathrooms, two of which don't have a sink. Didn't they know then that servants' dirty hands is the reason that typhoid spread?

But I would probably take this apartment anyway

I think there was a code change after the war. In almost all prewar apartments that had them, servants' bathrooms would have a tub and toilet but no sink. Each maid's room would have its own sink. I guess those toilet-only compartments at 834 would be an extension of that.

Then there're all the no-maids-room prewars with a toilet-only compartment off the kitchen.