The number of the Grover's diffuser iterations is proportional to square root of the amount of input data, but how is this realized when building a real quantum circuit? Is it simply by concatenating the needed number of diffusers one after another (in which case I assume the circuit designer is required to know the needed number of iterations beforehand)?
A quantum logic circuit isn't a literal physical object, it's a plan for what operations to perform. It's like sheet music. The piano maker doesn't have to know how long the song is when making the piano, they instead provide reusable keys that the player will just keep hitting until the song is done.
In a superconducting qubit device, the "keys" are microwave pulses bounced off the superconducting circuit. Each pulse performs some fixed operation. You would run Grover's algorithm by decomposing its operations into those pulses and playing them until the song is done.