Implementation of Leighton-Micali Hash-Based Signatures
It is stated, that LMOTS private key generation creates a 32B SEED at the start of the process, although does it create a new SEED for every new key from 0 to q=2^h-1?

If you use the suggested key generation method found in Appendix A, you use the same seed value for each Merkle tree.  If you use multiple Merkle trees (in HSS with L&gt;1), then each tree gets its own SEED value

although it is not stated for what q the LMOTS public key is generated, or does it mean, that this public key is the LMS public key, which is the root of the tree created by leaves of LMOTS public keys K?

It is the HSS public key, which is the LMS public key of the top level Merkle tree.

I have a slight problem using the reference implementation, as it is created for HSS. Do you know how can I force it to use only LMS?

Tell it to use L=1; that is, when you call hss_generate_private_key, pass it a 1 as the levels parameter (the second one).