You can, but I would be careful doing so, you don't want to be taking too many medications. DKnight is right when she says the z-pack is an antibiotic and it fights the bacteria, so it won't affect the DayQuil. If possible I would suggest staying home for the day and just resting as much as possible, this is really what is going to make you feel better.
Yes you can because the z-pack is an antibiotic and it only fights the bacteria, it does not treat the symptoms of your illness. The dayquil will treat the symptoms and help you feel better while you wait for the z-pack to fight enough of the infection to make you better. I have done this plenty of times since I tend to get bronchitis a lot.