Is there a proof that addition is faster than multiplication? The best upper bound known on the time complexity of multiplication is Martin Fürer's bound $n\log n2^{O(\log^* n)}$, which is more than linear time complexity of addition.  Do we have a proof that addition is inherently easier than multiplication?
No.

No unconditional better lower bound than the trivial $\Omega(n)$ is currently known for integer multiplication. There are some conditional lower bounds though. For more on this, you can have a look at Martin Fürer's paper Faster Integer Multiplication.

Edit following Andrej's comment: Addition can be done in time $\mathcal O(n)$. In comparison, the best known upper bound for multiplication is (approximately) $\mathcal O(n\log n)$. On the other hand, no non trivial lower bound is known for multiplication, thus there is no proof that addition is faster than multiplication yet. As (too) often in complexity theory, we just don't know!