Honestly going to a fully automatic process looks to me like a very complicated task.
I do not know how to reliable remove time from the images, apart from asking for the raw data, unannotated.
As for the diameter measurement, I would try to fit an ellipse on the dark contour of each cell. I don't think that there is a plugin existing that does that. I tried to do something like this years ago, and it did not come to fruition, so you would have to code yourself. To do so I would have a function that samples the perimeter of an ellipse, and minimize the pixel intensity of the pixels that are traversed by the perimeter.
But even this is not fully automated as you need to initialize the ellipse above each cell.