The difference in mass between KS and KL is very small. The reason is that the amplitude to go from K0 to anti-K0 is small. In fact:
the mass of K1 and K2 (which are to good approximation KS and KL) is given by the eigenvalues of the mass matrix M, where M11=M22=M and M21*=M12=δ.
The eigenvalues are calculated by taking the determinant of the matrix obtained by subtracting λ times the Identity from M.
This gives the equation:
λ²-2M*λ+(M²-δ²)=0
The difference between the solutions is 2*δ.

link

KS regeneration: it is due to the fact that K0 and anti-K0 have different quark content and so different nuclear cross sections for interaction with nucleons.
The K0 undergoes quasi-elastic scattering with nucleons(*), whereas its antiparticle can create hyperons.

(*) e.g. K0 p -> K+ n, since it's just a quark exchange between the kaon and the nucleon: (anti-s d)+(uud)->(anti-s u)+(udd).

talk on ε' measurements

There are two ways for a KL (almost CP=-1) to decay into 2π (CP=+1):

1. mixing (indirect CP violation): the ε fraction with CP=+1 decays into 2π; the relevant diagrams are boxes (ΔS=2)
2. direct CP violation: penguin diagram (ΔS=1) that goes from (sd) to (dd)

KLOE experiment:

Φ->KS+KL in 34% of cases.
KS and KL are in a pure antisymmetric superposition, due to the conservation of C in the strong decay Φ->K0+antiK0.
This provides a natural tagging based on lifetime. The KS and KL fiducial volumes are neatly separated.
Once one detects a KS, the opposite hemisphere contains a KL with 100% probability, and viceversa. Contaminations are at the level of 10^-5.

In principle one can look for CP violation by counting how many times KS->3π0, but in this case there is a further suppression due to the phase space. So, it's much easier to count how many times KL->2π0.