I have run TS12 and T:ANE from external hard drives, but those are the plug-in powered USB 3.0 hard drives. It worked quite well though a bit slower than an internal drive. A USB stick, sadly doesn't have the longevity and will die much sooner because unlike a hard drive is more like an SSD. USB sticks also get slower over time as they fill up and can only be reformatted, and unlike a regular SSD can't even be trimmed.
I have also run T:ANE over a network. I have the executable on my laptop and the data on my desktop. I shared the data drive and mapped a network drive to it on the laptop. I then configured the database location in T:ANE and everything hooked up. It wasn't too bad once everything loaded up, and it was quite nice sitting on my deck with my laptop using T:ANE instead of at my desktop on the nice day. The other advantage too is being able to connect to my data without needing to copy the contents to an external device, then copying routes and stuff back when done.