As Stephen writes, LP to SDP is trivial, just put the constraints on the diagonal (here done in primal space, an alternative is to parameterize the dual SDP). The SDP to LP is typically impossible, unless the SDP constraint is diagonal (or can be transformed to diagonal). If it would be...
K.s is used for semidefinite constraints and K.q are used for the SOCP cones. The data for the SOCP cones are stacked above the vectorized data for the SDPs.
If this sounds like jibberish to you, use a modelling language instead, such as YALMIP, and you don't have to think about these details...