And the church pic was so noisy because flash isn’t allowed inside the cathedral.
Noise is in all electrical devices. A weak signal is close to the noise so the noise becomes apparent. A strong signal is above the noise and appears clear. ISO is the gain control of an amplifier. Just like with stereo systems, you want a good signal to amplify and you want a very clean front-end amplifier so it is not introducing noise that is amplified by subsequent stages. Physically larger sensors provide a better signal and usually more expensive cameras have better, lower noise amplifiers. Those are the building blocks that go into the file your camera produces.
A lot of effort has gone into raw processing software. Adobe Camera Raw has pretty impressive noise suppression. DXO may have even better noise suppression. This was shot with a Canon Rebel T2i/550D at ISO 12,800, then double processed with DXO, once for the scene and once for the screen, then the results were combined in Photoshop to get the final image.
This was shot at 1/2oth, instead of 1/30th, so there is a little more brightness, a little more signal. It seems that makes it easier to get a clean result. Of course, any movement blurs at those slow shutter speeds. With better gear, you can get better numbers but there may be less depth of field. A 5D Mk III or a 1Dx can deliver a stop or two more ISO, as can a Nikon D4s, and fast primes at f/1.4 can deliver 4 stops more light than the f/5.6 I had available. Even in the crummy light, better gear could deliver 1/250th or even 1/500th with the same file quality the Rebel produced at 1/30th. That might give some latitude to back off on ISO to get less noise or to stop down a little to get better depth and sharpness.
If you have raw files, software may be a solution to a cleaner image. Processing out noise is an interesting exercise in a geeky sort of way. It is time consuming though. This level of noise reduction takes about 7 minutes on a really high end processor, for each pass.
If you are shooting architecture, you usually have time on your side. You can shoot at low ISO, even with a slow lens or smaller aperture. Exposure time will be longer, so you have to use a tripod, a clamp, or some furniture to provide a stable platform for the camera. Sometimes that provides an additional benefit. A really long shutter speed will remove moving people from the scene since they are not in one place long enough to register.