It's a bitch that while hard panned early stereo stuff sounds terrible on headphones, a lot of great music was recorded that way...look into crossfeed, which sends some of the L channel to the R and vice versa
I've been experimenting with it (via Qudelix K5 - there are also implementations for foobar) and am pretty happy with it so far
It has a very flexible (10 band graphic or 20 band parametric) EQ you can set however you like - so if you know or can figure out what exactly the Fiio EQ is doing you can surely come close
I'll get a weird "out of phase" sort of effect similar to what you're describing using Airpods with my (Android) phone sometimes (via BT of course) in the middle of listening once in a while - don't know what causes it, just have to dis/re-connect
It's "safe" as in being outdoors isn't going to harm them, it's potentially "unsafe" if you can't hear what's going on around you (traffic etc)
Tidal and Quboz should both have free 30 day trials so you can decide if you prefer them for yourself
(Personally, I still use Spotify)