wades....
IF it were me I'd start by narrowing down everything down. There are way too many unorthodoxies here to get to the root of the problem or "problems". FIRST unplug the one(s) going to the ROTEL. I'd also disconnect the one using "Analog Output function of our Sharp "
Get yourself down to one Apple TV or you'll never get to the bottom of this.
Further, I'd try outputting audio via Optical audio port, then using an optical audio(at ATV) to RCA (stereo analog) converter or cable.
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Some things to check:
Is the cabling (in particular any analog cables) touching other cables, electrical lines etc. (this is OFTEN the source of noise.)
Are there microwaves about (these will cause network interference).
Also, I'm curious how long you mean when you say "an extended period of time" How long are you referring to... I generally only run my ATV for a couple hours at a time, but I'm sure I've used it for 6-7 hours on occasion without issue.
To comment on your suspicions: Responding by number:
1. this could certainly be true (just like any piece of AV equipment) although I don't know about static coming as a result of of heat.. Are the units enclosed in a cabinet (and if so, is there a fair amount of space around them). The main reason I'd suspect this is that the problem doesn't exist until they've been on for awhile. But I wouldn't expect it to go away immediately on restart.
2. a bad DAC could certainly be a problem. (is there just one, that all are routed to? or do you have the same brand for each.
3. This easily be a copy-protection issue or an Amp issue.
COPY PROTECTION: As I assume you know HDMI was not designed for our convenience but as means of copy protection. In switching the HDMI audio signal to analog you are unintentionally attempting to override this protection. If any device in the chain doesn't like whats going on static would likely be the result. Further the software and it's multiple versions can conflict. This has not been as true of late but anything prior to v1.3 is suspect. The only reason NOT to suspect copy protection issues is that the problem doesn't rear it's head immediately. You mentioned iTunes downloads - these are copy protected. Try testing with some non-DRM vids: try some of the other av options in the menu.
Let me mention again that this issue only happens with one type of amp (the one that also switches video) and with purchased movies from itunes
Statement above would make me suspect copy-protection problems. If so, going optical audio -> analog might eliminate the issue.
AMP ISSUES: noise is a very commpon problem with amps, my sense is that they're more susceptible than other devices - but I'd expect noise issues to occur sooner (bear in mind I'm no pro on the issue the only time I had an issue: the culprit was cable TV interference) If it comes down to this, try the AVSforums - folks there would have good tips on noise canceling.
I'm afraid you've designed an [overly]complex setup. You really need to go back to square one with 1 ATV. Get that working then go from there.
Good luck.