Do restaurants like Botin or other nice restaurants require coat & tie for men? Also, tour books indicate a less casual dress code in Madrid all around, but in photos of Walks of Madrid & other sites, the people look pretty casually decked out. Any feedback, please? Thanks.

No, Botin and similar restaurants do not require coat & tie. You will only be required to wear coat and tie in a couple of very luxury restaurants, as Jockey, Horcher, el Bodegon, etc.