I have done a lot of investigation with regard to this problem, so I will try to summarize what I have learned:
1) Between the iPod connector and the Alpine IVA-NAV-10 "head unit" is the Alpine TUA-T550HD "HD Radio Tuner Box," which is unfortunately the weak link here. It is not a simple, direct cable like it should be. Due to the way that USB works, the HD Radio unit must act as a USB hub, and I suspect that this is where all of the problems lie. I'm guessing that the HD Radio shuts itself down when not in use, which tends to disconnect the iPod as well. Either that, or it's a really poor USB hub. Unfortunately, the HD Radio unit is hidden away in the Roadster, and you probably cannot access it without voiding your warranty. Your best bet is to wait until the Alpine system is fully started, then make sure your iOS device is awake when you plug it in. That usually works for me (*).
2) You do not need to reboot your iPod or iPhone. In general, it is a good idea to reboot your iOS device if you are having problems and have not done so in a very long time. I sometimes go for months without ever turning off my iPhone and have no issues, but then after a mysterious problem crops up I find that a quick reboot usually cures it. That said, the Alpine system fails too frequently for rebooting the iOS device to be a good idea.
3) You do not need to turn off Bluetooth, nor do you need to turn it on. Use Bluetooth only if you want your in-vehicle microphone and speaker system to work when placing or receiving calls. It's basically too noisy with the vehicle moving, but I leave it on anyway. It does not seem to affect the USB performance, although it may be acting as a red herring.
(*) We had a problem here in Seattle where my iPhone would not connect to my Alpine at all, but every other iPod and iPhone in the Tesla Store Seattle worked fine. Service was convinced that my iPhone had a problem, but as an iPhone developer and USB hardware developer, I know for a fact that my iPhone has absolutely no problems ever connecting with any other device on the first try. Sure enough, testing my iPhone on a new Roadster 2.5 on the showroom floor showed that my phone worked fine.
Upon ripping apart my car, and with my investigative help, we discovered that when wired directly to the head unit using the Alpine-supplied iPod cable, my iPhone works every time.
Thus the eventual "solution" was to replace the HD Radio unit with a new one, since it lies in between the iPod connector and the head unit. This seemed to work, in that I went from only having one day where it worked to having it work every trip if I'm persistent, but it's still far from 100% reliable.
I am fully tempted to remove the HD Radio unit so that the iPod connector would then be wired directly to the head unit. The direct link seemed 100% reliable, although I didn't test it more than the two days my Roadster was in the service bay. The thing is that HD Radio sounds like crap. If it required a paid subscription like the Sirius XM Satellite Radio system then I would rip it out immediately. At least for now, there's the conflicting temptation to access crappy-sounding music for free. I am also interested in maintaining my Tesla Motors warranty, so I won't be changing anything on the Infotainment system until the warranty has expired. Perhaps Tesla Motors charged $4,500 so they could afford to replace the cheap components frequently.