I always transfer prints directly from second fix to toner and always have.
Staining will occur if you do this if 1) the fixation of the print is not adequate and there are non-water-soluble silver halides left in the emulsion or 2) if the fixer is too acidic. FWIW, I use Ilford Rapid Fixer or Hypam 1+9 and have never had a problem. I've also used TF-4 and Kodak Rapid Fix (print dilution). Problems with staining may occur with the stronger "film dilution" of Ilford and Kodak rapid fixers; I've never tried it. Bottom line, fix well and use a near-neutral or alkaline fixer. Or wash thoroughly before toning. That works too, it's just more time consuming.
The wash-aid step should be after fixing and a five-minute water rinse. The five-minute rinse before the wash aid is important and helps speed up washing according to Ilford. Mixing selenium toner with wash aid is wasteful if you toss the toner when the wash-aid is exhausted. If you keep the toner longer, then you don't have a wash-aid step at all. Better to do it separately and after the five-minute rinse.
I've written a lot here and over on the LF fora about keeping and replenishing selenium toner. I have two batches, one stronger, one weaker, that have been going for 10 years now. Toning is done by visual inspections and prints are removed from the toner when the desired tonal shift has been reached. When times get too slow, more stock toner solution is added to speed it back up. Amounts are not specific. Toner used this way needs to be filtered before and after use. I use regular coffee filter cones. Sometimes a batch will grow bacterial slime. If that's the case, it's better to toss it and mix fresh. I've only had that happen a time or two.
Best,
Doremus