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What to expect on Humphreys Peak near the end of March, both weather wise and surface wise? So far looking at forecast I have an impression that it would be winter conditions, but does anyone have real experience there? Would I need microspikes to summit?

I summited Humphreys Pk & Agassiz Pk on 30MAR last year.

The southwest is having a very, very dry year due to La Niña weather pattern, but they are getting more snow as of late. Both peaks sit at the top of the Snowbowl ski resort, so if you check webcams you may get a more accurate idea of coverage. I experienced what would be considered mid-Spring conditions in Colorado, with a lot of wind. Snowshoes and microspikes were mandatory for my day.

And from recollection, Agassi peak is off limits unless there is snow on it.

That is still the case as far as I know. The closure is to protect a certain species flower and the restriction applies to the entire Kachina Peaks Wilderness Area above 11,400, no off trail hiking/camping. I have never been there but we thought about doing these last November until I couldn't take as many days off as needed so we just went to Bryce in Utah. Anyway, from our research you are allowed to hike on snow, but I struggled to find that in writing from the Forest Service. While supposedly legal to do with snow, they don't seem to advertise or encourage that fact so you may want to verify with the ranger if you are worried about the $500 fine.

24 inches predicted for Snowbowl ski area tomorrow.

Should be ok climbing Agassiz then! Monitor any avy risk though...