So basically what I'm asking is: will downloading files (ie., video files, DOC files ect) compromise my anonomity? I ask because I visited some forums selling a answer to this very question and some people are saying that it is NOT safe to download files of any kind though TOR. I don't believe they only mean that the file(s) will "call home". It seems to me that they are saying that simply downloading files via TOR can compromise your anonymity (even if the file doesn't "call home"). Can anyone confirm or deny this? Thanks in advance!
Yes, it is safe to download files.
However, it is almost never safe to open them outside of Tor Browser or a special environment.
Even non-executable files (documents, videos, audio files, etc) can be used to ping-back to some server on the internet and since the application opening them isn't going through Tor, you will connect outside of the Tor network.
This is a very old tactic of bugging files (sometimes government documents have such tracking in them, provided by a DLP contractor), so that if they are opened they can discover who opened them.
So, it is safe to download file over Tor, at least about as safe as downloading them on a shared or public wireless network, for example but it is definitely not safe to open them and it will harm your anonymity.
This presentation from 2015 shows a few examples of such techniques, although it is not at all extensive. There are plenty it doesn't cover either. These methods have been successfully used to deanonymize Tor users before, as well as deanonymizing state sponsored hackers targeting journalists.
The Tor Browser actually has an explicit warning about this before you download any files.
Using a Qubes Disposable VM, Tails or Whonix to open files after you've downloaded them will help mitigate many of the risks (against leaks through normal operation of the file type) but not always against exploitation.
I'd say in general there is no additional risk compared to regular use. After all, downloading files is what Tor Browser Bundle does all the time.
However, if you download something out of the ordinary, say a really big file or a lot of files in a specific pattern, it might be possible to correlate the traffic from the server to the Tor network and the traffic from the network to you. This requires both your ISP's logs and the download server ISP's logs to be compared. Or having a dystopian government, monitoring every connection, possibly colluding with like-minded governments. Or being a moron.
Tor is basically the same as hiding in a crowd. So if you avoid doing stuff to stand out and if the crowd is big enough, downloading files should be fine.