What happens when I call std::mem::drop with a reference
The reference itself is dropped.
a reference instead of an owned value
A reference is a value.
Why am I still able to use k after dropping it?
Because immutable pointers implement Copy. You pass in a copy of the reference and it's dropped.
Does drop not deref a reference automatically?
No, it does not.
what does the implementation of Drop look like for &str? 
There isn't one for any kind of reference, immutable or mutable, so it's effectively 1:
impl Drop for &str {
    fn drop(&mut self) {}
}
See also:
1 — As Peter Hall points out, there is a difference between having an empty Drop implementation and having no user-provided Drop implementation, but for the purposes of this question they are the same.