First of all, let me say that there is nothing perculiar about 3d HS theories - I mean the Vasiliev equations look exactly the same as in any other dimension.
The starting point in HS theory is to gauge HS algebra, so the generalized spin-connection and generalized dreibeins are just particular components of a single connection of a HS algebra. Connection of a HS algebra is just usual Yang-Mills connection, but the algebra is not $su(n)$ but something usually infinite-dimensional.
Some explanation comes from gravity which, as is known, can be reformulated as a gauge theory of Poincare, de Sitter or anti-de Sitter algebra. Again, vielbein and spin-connection are particular components of a connection of one of the above mentioned algebras.
HS algebras are infinite-dimensional (I ignore $sl(n)$ that was considered by many people in $3d$ since it is not a part of a consistent HS theory --- one can take $sl(n)$ Chern-Simons, of course, but it is not a full HS theor). Hence the space they act geometrically is infinite-dimensional as well and because of that it is not a well-studied topic.
Therefore, to both of your questions I would answer no and no.