  We study the two-impurity Kondo model (TIKM) in two dimensions with
spin-orbit coupled conduction electrons. In the first part of the paper we
analyze how spin-orbit interactions of Rashba as well as Dresselhaus type
influence the Kondo and RKKY interactions in the TIKM, generalizing results
obtained by H. Imamura {\em et al.} (2004) and J. Malecki (2007). Using our
findings we then explore the effect from spin-orbit interactions on the
non-Fermi liquid quantum critical transition between the RKKY-singlet and
Kondo-screened RKKY-triplet states. We argue that spin-orbit interactions under
certain conditions produce a line of critical points exhibiting the same
leading scaling behavior as that of the ordinary TIKM. In the second part of
the paper we shift focus and turn to the question of how spin-orbit
interactions affect the entanglement between two localized RKKY-coupled spins
in the parameter regime where the competition from the direct Kondo interaction
can be neglected. Using data for a device with two spinful quantum dots
patterned in a gated InAs heterostructure we show that a gate-controlled
spin-orbit interaction may drive a maximally entangled state to one with
vanishing entanglement, or vice versa (as measured by the concurrence). This
has important implications for proposals using RKKY interactions for nonlocal
control of qubit entanglement in semiconductor heterostructures.
