Yes, so long as the frosting has set properly. (If the frosting is very soft, the spray may drip/run into the frosting giving an odd appearance).
It is not clear whether the question refers to a box cake mix still in the package, or a cake made from a boxed mix, that has been frosted. In that case, the answer depends on the type of frosting. A cake prepared from a mix, frosted with buttercream type icing should be fine for three days, and longer, as long as it is kept covered and cool. It does not need refrigerated unless the room temperature is quite warm.
The time required for buttercream to develop a firm crust depends on a number of factors, including the proportion of ingredients used, the temperature of the ingredients when combined, and the temperature and humidity of the room. Generally, if you want to be certain the buttercream crusts over fully, you should leave the cake uncovered overnight. It will crust more quickly in the refrigerator, but will still take several hours. > It would take several days for buttercream frosting to become completely hard all the way through. Even then it will be quite soft. Hard icing decorations are not made out of buttercream but out of royal icing.
It starts to melt!
The same lengh of time as the use-by date on the cream cheese, provided the frosting is kept in the refridgerator. (Assuming all the other ingredients have a use-by date that is later than the one on the cream cheese. If any expire earlier, use the frosting by the earliest use by date of any of the ingredients).
it depends on temperature and relative humidity of the storage
about a 3 days to a week
it can stay fresh for 2 weeks
So long as it was kept at room temperature, it should be fine to eat after it was left out overnight.
If you want my personal thought, which may not be correct, I think maybe in the next day it will ROT and stink. If it's a type of cheese, yeah, it will stink and rot. Cream cheese should not sit out at room temperature for more than 2 hours. Less time is better.
Bacteria can form on food left out of the refrigerator after an hour.
If you are smart, you probably won't.