Can allergies cause foul smell and taste?

Yes. If you have significant inflammation in the sinuses, there can be residual changes in the sense of smell and taste. Additionally, if you make copious amounts of mucus, then bad breath is very common.
Yes. Some people who are chronically congested from their allergies will also get chronic sinus infections. These people often have a foul smell and taste that is long-lasting. Aggressive management of their allergies often improves their smell, taste and sinus problems.