Are regular languages closed under addition?
Yes, they are.

First, consider the alphabet $\Sigma_i^3$ whose symbols are triples of digits (stacked one above each other into a pile of three digits). Over this alphabet, we can define a regular language $A'$ where the string formed by the topmost of the three digits belongs to $A$, a regular language $B'$ where the string formed by middle of the three digits belongs to $B$, and a regular language $C$ where the top two strings sum to the bottom one. $A'$ and $B'$ just use modified automata for $A$ and $B$, while $C$ uses the fact that you can do addition by scanning right-to-left while keeping only a single digit of carry as state.

Then $A'\cap B'\cap C$ is (by closure under intersection) a regular language that recognizes stacks of three strings, one in $A$, one in $B$, and the third in the sum. The homomorphism that strips off the top two digits from a stack leaving only the bottom one takes this to the language you want, and the result follows by closure under homomorphism.