evb, you ask a lot of questions! But they're good questions, so here goes.
(a) Common failure modes. None that I've heard of. Since these things are NOS, there might be some infant mortality, but I've heard that the seller is quite good about replacements.
(b) Delicate equipment? Hell, no! They were made by HP for Lucent. It doesn't get much better than that. I have an older member of the family. I've had it for a few years and it's been monitored continuously since I got it. I suspect that most of the units in service with hobbyists are the same. There was a question about whether the Z38xx program had a subtle bug that could wear out the EEPROM in the Z3801A and, by extension, other members of the family. The author patched the program to remove that danger.
(c) Dying for no reason. No, that was a somewhat off-topic comment about a totally unrelated unit (Rockwell Jupiter GPS receiver).
(d) Data collection. Start with the program called Z38xx written by the late Ulrich Bangert. Although it needs a hack to recognize these units, it's a minor thing. If that program isn't to your taste, roll your own. It's just a serial port.
(e) Antennas. Get a timing grade antenna. Typically, they're bullet shaped rather than a puck. The timing grade antennas have better filtering and typically higher gain than other types. Make sure it will tolerate 5V. Newer ones might be 3V3 only. More definitive recommendations are difficult because local conditions like cable length and local signal obstructions will have an effect. Earlier in the thread motocoder and I posted graphs and info on our setups. That's as good a guide as any.
(f) and (g) Initial vs. burned-in performance. That's not clear. I'm beginning to wonder just what their ultimate performance will be. As we speak, I'm playing with yet another GPSDO from Trimble. The performance is quite similar to the Lucent boxes we're discussing here. But when I removed the OCXO from the board and measured it seperately, its short-term performance was much better than its short-term performance while on the board. It seems that the board is reducing the OCXOs short-term performance while improving its long term performance. There's no way to tell if that was intentional or irrelevent to the designers. So until we hear from someone who has had these units online for a few months - and there should be some since they first came on the market late last year - we won't know what the ultimate performance will be.
But remember, I'm coming at it from the point of view of a slightly obsessed, PITA, time-nut who bitches and whines about a GPS receiver that has a 10 MHz output that's off by 0.00015 Hz! For sane, normal people these units will perform perfectly well out of the box.
(h) Run one unit only? There was discussion on that, but since I got these as spares, I haven't investigated. You should be able to find the info on time-nuts.
(i) Reaction to moving a unit? Anytime you move the antenna, you introduce some error into the system. The error can show up as more jitter in the output or as jumps when a satellite joins or leaves the list of tracked birds. Whether the error is significant will depend on your application. In general, when you move the antenna, you should rerun the position survey. I think there's a command for that, but I'm not sure.
Ed