Web addition will work fine for Cyclone III. However NIOS is a licensed core.
So if you have NIOS, it will require a NIOS add on license. to build it without the timeout. I've never have done NIOS on a web-license (we have a full license with the NIOS add-on)
If you go Cyclone V, it appears all the SOC's are supported under the web edition. So that would be my recommend place to start with a CPU.
Basically the smaller (Current) families are supported on the web-edition. The license gets you support for larger family members and some optimizations and tools not supported in web-edition. NIOS is an add-on license, but a "Time-out" version is available to try before you buy.
Pete