Based on my decoction experience (haven't done non-direct fired step mashes), you do need to stir mash when adding hot water, but I don't think you need to stir the crap out of it. Just a good stir to get the hot water to distribute throughout the mash. Hot always moves towards cold, so the mash would eventually come to an equilibrium temperature even if you didn't stir... you're just trying to expedite that and prevent hot spots from sitting too long.
I'm not sure about your temp steps though. You might consider low 130s to mid/upper 150s, then to mash-out. That 122 step might reduce head retention depending on your malt. Unless you have the malt analysis to indicate otherwise, I would assume it to have a higher Soluble Nitrogen Ratio than is good for a 122 step.