Although the answer is already accepted, I'll add an answer with a different look on things.
1. Is it safe to just use SHA-512 as the hashing algorithm to get a 512-bit output and split it in 2?
It's safe, but that's not what HKDF-expand should be used for. The idea of HKDF-expand is to call it twice, once for each key, using the
info element to distinguish between the two (e.g.
info could be an ASCII encoding of the key name). Note that HKDF should have a relatively short running time.
2. I know that using salt is not mandatory, but does using it actually improve security?
Although it is not required, it is best to use a salt. So unless there is some important reason not to - such as the requirement for a deterministic protocol or the unavailability of a random number generator, please do so. It is relatively unusual for a KBKDF algorithm to have a salt parameter, but the authors seem to think it is important; see the quote underneath this answer.
3. I know this might be a dumb idea, but still ... can the salt be the IV used for encryption?
You could, however I would urge you to independently generate it using HKDF-expand, see answer (1).
Quote for part #2 (see 3.1 of the RFC):
We stress, however, that the use of salt adds significantly to the
strength of HKDF, ensuring independence between different uses of the
hash function, supporting "source-independent" extraction, and
strengthening the analytical results that back the HKDF design.