https://1drv.ms/x/s!AldNdi8CoTHPkT_1-3iZhT3dHamE?e=HYuOL3
Thank you.
There is an issue with the date formula, you compare if E3 is lower today, that is also true if the cell is empty, because an empty cell is calculated as 0 and therefore this condition is always true.
The condition if C3 is empty is IMHO superfluous.
And if you want to colors if there is no patient we have to apply a condition as Hans suggested, but we have to stop the formatting:

As you see I've changed some formulas, so here is your file, take a look if it works as desired.
https://www.dropbox.com/s/n2mwbdi08w25az0/631e1a89-fee2-4188-a82b-430574f54dcb.xlsx?dl=1
Andreas.