Does Arduino have separate sample-and-hold circuitry for each analog input pin
No. The channels are multiplexed. I wonder if you might have better luck with an external A/D converter.
After thinking about it for a little while, I realized that my application (a phased array of microphones, with the processing done on the PC-side) doesn't require simultaneous sampling, as long as I know which analog input is being sampled and at what time. I would still have to interpolate between samples when combining the signals even if simultaneous sampling were possible.
And I've looked at different A/D chips and acquisition interfaces. No commercially sold and packaged DAQ device comes close to an Arduino (IF I can sample 8 analog signals at least at 8-bits and, say, 8kHz each... while streaming to my PC at almost 2Mbit/s) or some similar kind of microcontroller for a similar price. Since the Arduino seems to have such a rich community and because I like the relatively-easy-learn environment and because it's so cheap, I think it's a pretty good fit. I hope to later do some similar work with ultrasound in the >1MHz range, but for that I will need a different microcontroller (or, actually, many microcontrollers, with the data captured initially onto their on-board SRAM, since I don't think I'll be able to stream that much data continuously at full duty cycle), since the typical Arduino-type microcontrollers don't support that many samples per second.
If I had thousands of dollars to spend, I would just buy myself some NI high-speed DAQ equipment and LabView (that's what I did before... I'm trying to replicate my senior thesis project using equipment that I own instead of my alma mader's equipment which I no longer have access to). But I don't have much of a disposable income right now.
Eventually, though, I'd like to get into the >100MHz range so I can do some phased array imaging with radiowaves (underground imaging for amateur archeology). But that is clearly out of the realm of microcontrollers. I will need either a different approach or more money for that. Perhaps an arduino driving some programmable analog delay lines.
DAQ equipment (and software like Labview) is expensive. Arduino and associated software are far cheaper.