Q. If one happens to have some server-grade hardware at ones disposal, is it ever advisable to run ZFS on top of a hardware-based RAID1 or some such?
A. It is strongly preferable to run ZFS straight to disk, and not make use of any form of RAID in between. Whether or not a system that effectively requires you make use of the RAID card precludes the use of ZFS has more to do with the OTHER benefits of ZFS than it does data resiliency. Flat out, if there's an underlying RAID card responsible for providing a single LUN to ZFS, ZFS is not going to improve data resiliency. If your only reason for going with ZFS in the first place is data resiliency improvement, then you just lost all reason for using it. However, ZFS also provides ARC/L2ARC, compression, snapshots, clones, and various other improvements that you might also want, and in that case, perhaps it is still your filesystem of choice.
Q. Should one turn off the hardware-based RAID, and run ZFS on a mirror or a raidz zpool instead?
A. Yes, if at all possible. Some RAID cards allow pass-through mode. If it has it, this is the preferable thing to do.
Q. With the hardware RAID functionality turned off, are hardware-RAID-based SATA2 and SAS controllers more or less likely to hide read and write errors than non-hardware-RAID controllers would?
A. This is entirely dependent on the RAID card in question. You'll have to pore over the manual or contact the manufacturer/vendor of the RAID card to find out. Some very much do, yes, especially if 'turning off' the RAID functionality doesn't actually completely turn it off.
Q. In terms of non-customisable servers, if one has a situation where a hardware RAID controller is effectively cost-neutral (or even lowers the cost of the pre-built server offering, since its presence improves the likelihood of the hosting company providing complementary IPMI access), should it at all be avoided? But should it be sought after?
A. This is much the same question as your first one. Again - if your only desire to use ZFS is an improvement in data resiliency, and your chosen hardware platform requires a RAID card provide a single LUN to ZFS (or multiple LUN's, but you have ZFS stripe across them), then you're doing nothing to improve data resiliency and thus your choice of ZFS may not be appropriate. If, however, you find any of the other ZFS features useful, it may still be.
I do want to add an additional concern - the above answers rely on the idea that the use of a hardware RAID card underneath ZFS does nothing to harm ZFS beyond removing its ability to improve data resiliency. The truth is that's more of a gray area. There are various tuneables and assumptions within ZFS that don't necessarily operate as well when handed multi-disk LUN's instead of raw disks. Most of this can be negated with proper tuning, but out of the box, you won't be as efficient on ZFS on top of large RAID LUN's as you would have been on top of individual spindles.
Further, there's some evidence to suggest that the very different manner in which ZFS talks to LUN's as opposed to more traditional filesystems often invokes code paths in the RAID controller and workloads that they're not as used to, which can lead to oddities. Most notably, you'll probably be doing yourself a favor by disabling the ZIL functionality entirely on any pool you place on top of a single LUN if you're not also providing a separate log device, though of course I'd highly recommend you DO provide the pool a separate raw log device (that isn't a LUN from the RAID card, if at all possible).