Orientation puzzle(s)

Let me begin with a fill-in-the-gap puzzle.

C× ~ S^1	BS1 aka K(Z,2) aka CP∞	complex orientability of a homology theory: h(CP∞)=h(pt)[[u]] (deg u=2); example: H(-;Z)	complex vector bundles, Chern classes	complex K-theory, BU	unitary cobordism, MU
R× ~ Z/2Z	B(Z/2Z) aka K(Z/2,1) aka RP∞	real orientability of a homology theory: h(RP∞)=h(pt)[[t]] (deg t=1); example: H(-;Z/2Z)	real vector bundles, Stiefel-Whitney classes	real K-theory, BO	unoriented cobordism, MO
Z/pZ (p>2)	B(Z/pZ) aka K(Z/pZ,1) aka L_p	“Z/pZ-orientability” of a homology theory: h(L_p)=h(pt)[[ε,u]] (deg ε=1, deg u=2; &epsilon2=0); example: H(-;Z/pZ)	???	???	??? (universal Z/pZ-oriented theory)

Well, at least in the complex case (the first line of the table) it's known how to construct corresponding homology theories starting from the classifying space: consider Σ^∞CP^∞ and invert a generator of π₂CP^∞; by Snaith's (?) theorem resulting space is homotopy equivalent to BU. And applying the same procedure one more time yields unitary cobordism: Σ^∞BU[1/β]=\wedge_nMU(n). I wonder, if there is a variant of that construction which works in the real case? N.B. inverting generator of π₁ in Σ^∞RP^∞ or Σ^∞BO yields a contractible space.

And the last question (in this post, I mean :-). Consider R:=H^*(B(Z/p);Z/p)=Z/pZ[[ε,u]]. Structure of H-space on B(Z/p) induces on Spec(R) structure of an algebraic group — and in fact it's just A^1|1 (a topologist would probably say something like "structure of 1|1-dimensional additive formal group on Z/pZ"). Theorem: Z/pZ[Aut Spec(R)]=A_p, where A_p is the automorphisms algebra of the homology functor (a topologist would say "algebra of mod p cohomological operations") aka mod p Steenrod algebra. Does anyone know an explanation of this theorem?