As for question 1, to use Solve it is better to use exact coefficients:
\[CapitalGamma] = 1 ; L = 1/100;
OD = 100; \[Gamma] = 0; \[CapitalDelta]pp = -100 \[CapitalGamma]; \
\[CapitalDelta]c = 0 \[CapitalGamma]; \[CapitalOmega]pp =
3 \[CapitalGamma]; \[CapitalOmega]c =
1 \[CapitalGamma]; \[CapitalOmega]p0 =
1/10 \[CapitalGamma]; \[CapitalOmega]s0 = 0 \[CapitalGamma];
\[Gamma]21 = \[Gamma]*\[CapitalGamma]; \[Gamma]31 = \[Gamma]41 = \
\[Gamma]32 = \[Gamma]42 = \[Gamma]43 = \[CapitalGamma]/2;
Eta = (OD \[CapitalGamma])/(2 L);
A\[CapitalOmega]p = 1; A\[CapitalOmega]s = 0;
{sol07} = Solve[...
This way I get an exact solution.