All the answers can be found in the original paper: "Cryptographic Extraction and Key Derivation: The HKDF Scheme" which is generally available.
Which properties are fullfilled, when IKM and the salt are random and secret, but not uniformly random? Are the output keys uniformly random?
They should be close to uniformly random yes, as the output of SHA-X is supposed to be uniformly random, assuming that the input isn't chosen deliberately.
The salt is generally not considered to be secret, although it can be.
Which guarantees does HKDF give, if (a.) IKM or (b.) salt got leaked?
If you leak both then you'd leak the output as well of course. As indicated, the salt is usually considered public. If you leak the IKM then the algorithm is deterministic, given a known salt, so an adversary can calculate the OKM.
Is it also uniformly random, if no salt is provided at all?
Again, unless the IKM is deliberately taken from a set known to produce specific results given a salt. The salt does make sure that domain separation takes place.
Furthermore, is there any paper that clearly provides answers to this question?
We can take this text from the HKDF paper:
Informally, a randomness extractor is a family of functions indexed by a public, i.e., nonsecret,
parameter (which we refer to as "salt") with the property that on any input distribution
with sufficiently high entropy, if one chooses a salt value at random (and independently of the
source distribution) the output of the extractor is statistically close to uniform (see below for a
formal definition). Moreover, this statistical closeness holds even if conditioned on the salt value.
