Noisy and strange bird behaviour (blackbird?)

Hello everyone! 

For a few days now my garden has been inhabited by some very annoying birds. I have been trying to identify them but based on the bird calls that I can find in the bird finder here, I can't quite make out what they are. They look like blackbirds (although they're brown) but they don't sound like the recording! This bird's call is very repetitive, very loud and extremely annoying. The bird itself I think is building a nest in a tree in my garden - the reason why I can't quite grasp, since I have a cat, and the bird has definitely seen him! So one bird hops around the garden with this noisy incessant call (it really goes on all day), but I have also seen other more silent birds of the same species around. I went out to investigate, and these birds were just sitting still, blinking their eyes but not moving an inch. It was incredible. My cat went close as well and still they didn't move (the cat was very confused and just walked away as a consequence). 

Based on the description, it would definitely seem like a blackbird (female). I don't understand the behaviour though - surely if they identify a predator they should move away? There's plenty of trees around and not as many cats - my garden seems an odd choice. I also have a constant headache now as it's just like a car alarm and there's no getting them to fly away. I approach and wave my hands/make noises, and they stay still. Is there a solution? What bigger deterrent could there be than a cat?!