This doesn't look like a fixer problem.
That said, it is possible to get fogging in the fix if you turn on white lights prematurely. I've never managed to do it, however, even by turning on the lights with prints in the fix <30 seconds.
It would be more likely to occur if you're stop bath is spent and you are using an alkaline fix, or if the fix is spent. Safelight fogging in the fixer would be extremely unlikely.

Some papers could require more time, it's hard to determine without doing tests for retained silver in the paper. Most manufactures have recommended fix times, and if you work according to those requirements with unspent fixer you should be fine.

Some papers have different safelight requirements too, for example Ilford papers are ok with amber safelights but some of the eastern European papers require red safelights.