Eosin, being yellow, is a sensitizer to blue light and is a weak sensitizer IIRC.
It will add no really noticable difference to the speed of the emulsion, nor any significant change to the spectral sensitivity, as the emulsion is already blue sensitive.
The sequence is outlined in another post in more detail, but basically you add the hypo, with or without gold, hold at a predetermined temperature for a predetermined time. I am using 1 hour at 60 degrees C. for my emulsions, but that may vary.
After this, you add the dye at 40 deg C and hold for 15 minutes. That is the usual sequence.
However, if it is premade by a manufacturer, there may already be dye on the emulsion, or it may have restrainers in it to prevent the action of the addition of hypo. If it is already treated with hypo, further treatment will fog it, in all probability.
I have posted elsewhere, that freezing an emulsion will ruin it in most all cases. It disrupts the action of the gelatin by denaturing it and will often cause very bad results.
PE