The answers to all your questions are basically yes.
If you are building a brand new wheel with new rim, hub, and spokes, you might want to buy a new wheel or even wheelset. It is more economically this way. There is a wheelset with Weinmann DA17 rims, SS spokes, Nexus-3 (non coaster brake) hub, tubes and 700x32 tires for less than $200.
If you are using an existing wheel and it has a high flange hub, then you can get an older 3-speed hub with similar flange diameter so you can use the same spokes. The older Sturmey Archer AW3 has a slightly larger flange diameter than the older Shimano (non Nexus) hubs. If you have a low flange hub in it now, try a spoke calculator and change the cross pattern (add a cross), etc, you might still be able to use the old spokes. This way, you might get a rear wheel built for less than $20-25 (for the cost of an older hub).
I have various both the older 3-speed hubs (AW and Shimano) and the newer Nexus, the older ones are just as good, if not better. The newer ones do look nicer, though.
If you are not against coaster brake, I would get a 3-speed coaster brake hub. It is really a combination of the best in both worlds. Just add a front rim brake. I have two such hubs (both older Shimano hubs) and one of them is on a Worksman Industrial (heavy duty) tricycle and it works very well, even under load.
Last edited by loubapache; 10-14-14 at 07:30 AM.