First, there are many similar questions here, but I found none with direct answers about this specific scenario. If you find a duplicate, please be kind enough to link to the specific answer(s). Also, to avoid question being too broad, consider this specific scenario only.
Assume a proprietary Linux application, which links against LGPL libraries dynamically, and does not link against GPL libraries. It also executes GPL'ed binaries from the file system through standard OS APIs though, to perform some actions.
Then assume a customized mainstream Linux distribution, with visible trademarked logos etc removed.
Then assume a self-contained product, which includes hardware, let's say Raspberry Pi with a bootable SD card with this customized OS. Then it has the proprietary software, which ultimately needs to start when device is switched on.
How to distribute this combination legally?
More specifically, I see (at least) three separate concerns:
In general, how to distribute just the customized Linux distro legally? For example, are there tools, or which distros have tools, to download all used default source packages, ready to be included with distribution or archived locally?
I think it's sometimes thought, that if you distribute a derivative of a mainstream Linux distro, you do not need to distribute the sources of default packages yourself, or that you can just point to the distro source repositories if someone asks for them, but I'm not sure if this is legally valid way to fulfill GPL obligations.
How to legally include the proprietary software with the distribution? For example, is it ok to have the proprietary software already installed in the above mentioned SD? Is it ok to have it uninstalled on the same media, and prompt user for it's installation on first boot? Is it required to have it on separate media or downloaded from the net, and installed from that? What are the limits of "certainly OK" and "certainly not OK", is there any consensus or even legal precedent?
Is executing GPL programs with hard-coded command strings from a proprietary application OK? My understanding is, this is OK and does not create a derived work. But if this is an issue, would working around it be as simple as creating an LGPL-licensed wrapper? Or just having the commands in modifiable configuration file (a good idea anyway, regardless of licensing issues)?
Note: Except for the techincal aspects of first bullet point, I understand that answers here are not any kind of legal counsel, but answers need to be backed up somehow, this site is not for opinion polls.