Are there rest rooms in the Cathedral?

No restrooms in the church. There is a public restroom in Olympia Tower in 51st street (across the church)

no

I ASKED AND THE MAN POINTED TO THE BUILDING IN FRONT OF ... SO I GUESS NO.

It's a church that holds regular services, so yes. I think they are towards the front on the right hand side, behind a section of great big wood doors.

Yes

Not sure, but would imagine so.