Well technically all starships have secondary engines Warp and Impulse. However having a back-up warp core does sound like an extremely good idea (in the real world), to me it runs a little against the Trek feel. One other fact that needs to be considered, redundant system damage. It's hard enough to keep one highly dangerous anti-matter engine from being damaged (i.e. Core Breech), let alone two engines.
Now as far as your Sick-bay question, I've always assumed you have smaller medical stations and triage centers at key locations through out a ship.
As to the third question - "what would public reaction be for a ship of this nature?" are we talking about here on the boards or in the eyes of the Federation Public?
Phoenix...
"I'm not saying there should be capital punishment for stupidity,
but maybe we should just remove all the safety lables and let nature take it's course"
"A Place For Everything & Nothing In It's Place"