I think the TJ08-E is a great case, specially for a compact build, if you wish to see some ideas check my signature i did the micromi build on it. The only downside imo is the frontal fan which at stock low settings is still a very noisy fan, but below 600rpm is tolerable, if FanXpert2 is all the good that its said, the mobo should be able to control it.
Now with your questions,
1. Is this fan/airflow layout going to work ok? Hopefully while keeping all fans running at low rpm?
It should, the PSU will be passive until needed, so i so no issues, in my build i used a X660 and until gaming its a passive psu, the 560 should be fine.
2. Will I be able to control the voltage and/or RPM of all these fans using the ASUS Maximus V Gene motherboard & Fan Xpert 2? Does this include the GPU and PSU fans?
I only own the Gene IV Z, and i cant control 3pin fans, but according to asus, the Gene V should be able to control 3pin and 4pin, but this is something that only an owner could confirm, so keep looking. According to asus it can control all fans... but i have my doubts if its going to manage all case fans as one, but this is something that again only an owner will be able to say. Now the GPU and PSU.... you will need to cable them out and plug them to the mobo, i wouldnt do it, you would void the warranty on the PSU by opening it, and not worth it, as it is the X560 is a very quiet PSU should be passive if no load is applied to the PC. With the GPU.... imo not worth it either, you will need to cable to you mobo, but there is no need, with MSI Afterbuner you can control most of the GPU fans, i would investigate a little more into getting a good GPU that its not that noisy and control it with MSI afterburner.
3. Temperature-wise - can I get away with just using one 92mm or 120mm fan on the HS? (i5 3570k CPU - not OCed now - I may in the future. It will be under load often though for Photoshop and/or gaming). Removing the top fan would lower the profile and maybe improve airflow?
4. Has anyone tried using this HS with two 120mm fans to improve sound compatibility?
Im not going to respond on this as i dont own that cooler, but personally i dont think its that great, its more a low profile cooler and the TJ08-E can fit very high coolers, you can check my build, i fitted a Thermalright HR02 (not the macho version), with this in mind i would go with Thermalright HR02 Macho, cheap and big, the fan on it is very good imo (i also used it on minimi build). If you dont have access to it where you live, my choice would be Scythe Mugen 3, it comes with Slipstream PWM 120mm fan that can be drop down to 250rpm, so its very quiet and has good performance, you can see my HTPCmi build for temps, and i would add another Scythe Slipstream PWM (SY1225SL12M-P) for the back case fan, i think this combo should work very well, specially because the GENE V has dual CPU fans, so you can connect the CPU fan and the back case fan to the CPU mobo connectors having an even flow.
5. Should I place the PSU fan-down (counter-intuitive and strange given the existing top vent in the case but recommended by Silverstone) or fan-up?
Its up to you, the X560 will be passive on idle, so since hot air raises, i would install it with the fan facing up, thats how mine is installed and no issues.
6. Will a 7,200rpm SG-Barracuda 3.5in HDD get too hot if it is not in the HDD cage directly in front of the intake fan? (the fan doesn't extend down to the bottom external bay or up to the 5.25 bays)
I have a 7200rpm samsung F3 500gb on the bottom cage with an ssd on the bottom, and its temps are fine, check my build i have some hwinfo with hdd temps on it. Now there is no grommets to lower the hdds noise, so my recommendations is to invest a little more on an SSD, like crucial m4 or samsung 830, and get a green 5200rpm drive like WD Green Caviar, Samsung F4 or Hitachi 5k3000 for storage.