Processing power is not a problem these days. Maybe if you feed all channels at ultrasound frequency to ADC and then try to do all processing in software. But you don't have to, it's oftentimes easier to have hardware (FPGA) pseudorandom signal generators and correlators to do the bulk of signal processing for you.
It depends. An FPGA able to do that is a big one, quite expensive. Today, an even more so tomorrow, you can afford to do that kind of computations in a CPU and/or a GPU.
Regardless of who is doing the beamforming, the TX and RX analog parts are intrinsically quite expensive with at least dozen of channels.
