What problems arise in gene expression when both introns AND exons are expressed?

Introns and exons are both present in eukaryotes, which are monocistronic. If the introns were translated as well, the polypeptide sequence would obviously be completely different - the required proteins would never be synthesised.

I they both would be expressed the resulting protein would not be of any use (unable to serve any useful purpose) so it would just spend some time in the cytoplasm before being digested by enzymes.
I don't have any particular source to site but if you are looking for a scientific way to account for it, I think you should look in the gene expression regulation topic! One you go through the steps of the regulation you will know on which of those steps that invalid protein will be destroyed but if you are looking for a particular study then most likely you won't find one :/ I don't think the proteins get expressed that way

It's not always bad. Intron mediated enhancement can be beneficial.