Decidability of equality, and soundness of expressions involving elementary arithmetic and exponentials
I don't know, but I suspect it's an open question.

If the theory of the reals with exponential function is decidable, then your problem is decidable, too.  It is known that if Shanuel's Conjecture holds, then the former is decidable, so your problem is too.

If I understand correctly, the following paper tackles your problem:

The identity problem for elementary functions and constants.  Dan Richardson, John Fitch.  ISSAC '94.

It appears that they show a procedure that always terminates if Shanuel's Conjecture holds; and if it doesn't terminate for a particular expression, then we can extract from that expression a counterexample to Shanuel's Conjecture.  They then argue that it seems unlikely that we will find a counterexample any time soon, so it seems unlikely we will find inputs on which this procedure fails to terminate any time soon.

See also https://mathoverflow.net/q/118972/37212 and https://mathoverflow.net/q/129563/37212 and https://mathoverflow.net/q/145299/37212.