This does not answer the question, since the question concerns both genera of residual and of non reduced curves, but addresses only the issue of residual genus formulas, and in the case of P^3. The formula is taken from Semple and Roth, page 91. But it may help.
Ok, as in Semple and Roth, pages 90-91, consider a space curve C of degree n and genus p, lying on a surface S of degree d. Then pass another surface S’ of degree e through C, so that it cuts further on S, another residual curve C’ of degree n’ and genus p’. (C’ is not necessarily connected, so p’ could be negative.)
Then we claim: 2(p-p’) = (n-n’)(d+e-4).
This follows from the two adjunction formulas for C and for C’, both taken say on S, and subtracted.
By the adjunction formula for the curve C on the surface S, the degree of the canonical bundle K of C, equals C.C + (d-4)n, where C.C is the intersection number on the surface S. Similarly the degree of the canonical bundle of C’ computed as a curve on the same surface S, equals C’.C’ + (d-4)n’.
Subtracting, the difference of the degrees of the two canonical bundles equals deg(K) - deg(K’) = 2(p-p’) = (d-4)(n-n’) + C.C - C’.C’.
Now since S intersects S’ in the union C+C’, I claim (C+C’).C = e.(n), and (C+C’).C’ = en’. Subtracting, and canceling, gives C.C - C’.C’ = e(n-n’). Substituting above, gives 2(p-p’) = (d-4)(n-n’) + e(n-n’) = (d+e-4)(n-n’). Of course also n+n’ = de.
For example, if C is two disjoint lines lying on a cubic surface S, and S’ is a quadric surface passing through both lines, then C’ is a quartic curve. Then n=2, n’=4, p = -1, d=3, e=2, so 2(-1 -p’) = -2, so p’ = 0, and C’ is rational. This is example (2) at the top of p. 93, of Semple and Roth.