Is it technically possible to get a different upper in .223 and put it on this lower? Does it require changing any of the lower's parts? Does it require a complete rebuild? Or would other uppers be incompatible?
No. You'll need an AR-15 lower to shoot the punier calibers. You can get uppers in other calibers based on the .308 cartridge, or even some of the short magnum calibers.
The .308 platform is NOT mil spec. That is, uppers and other parts are not interchangeable between all platforms. For instance, an Armalite AR-10 upper cannot be used on a DPMS 308 lower. The DPMS 308 lower is compatible with CMMG and JP Rifles components.
Here is the warning from JP Rifles about the compatibility issues between the different 308 platforms.
As already mentioned, if you want to shoot .223, then you need an AR-15 lower. However, .260 Rem can be used in a .308 lower setup. There are a lot of wildcat cartridges that can be used in a 308 style lower. You just need to do some googling to find them.