Good Morning J., I am sorry to say whatever other kids have in your day care your child will more then likely get it also. Same thing happens at school.
As for the rash, gee not sure but it could be from something at day care also, if she sleeps in a crib with sheets being washed in something she is sensitive to.
If she will let you but a couple of drops of nasal saline for infants in her nose then use the bulb and suck it out. Use a humidifier at night also. The coughing could be from all the mucus going down her throat. If you think about it when we get a cold we have a lot of drainage and if it continues to go down our throats we sometimes get a chest cough also.
Our nurse had use use a mixture of A&D ointment ( nice sized blob) with about a TBS of Maalox to put on rashes on Zane, it started clearing up even after the first application. He has eczema very badly in winter. You really have to mix it up a lot to get it blended, and it stays put so you don't have to keep reapplying constantly.
Put it in a snack bowl with a lid to keep it.
God Bless and I hope you get some wonderful advice from other Mama's out there.
Have a Wonderful Easter SONday
K. Nana of 5