K no requirement
B reflexive and symmetric
$S_4$ reflexive and transitive
$S_5$ reflexive, symmetric, and transitive
(See also: Semantics section of
Correct, both statements are true.
$1.$ Every logical truth in K (Kripke) is also a logical truth in ρ (reflexive).
Since K have no conditions, everything we can prove in K, we can also prove in T.
$2.$ Every sentence that is not a logical truth in S5 (ρ,σ,τ) is also not a logical truth in σ (symmetry).
Since $S_5$ has most conditions, if $S_5$ can't prove it, no one can prove it.
Is it true that these worlds are proper subsets of each other with all that implies
(or are there more complicated relationships)?
Yes, you can think their relation as sets, by definition $S_5$ model is also a K model, D model, T model, B model, $S_4$ model, for example if a formula hold in all K models, directly implies it's valid in all $S_5$ models. In another word $S_5$ models $\subset$ K models.
Here is system relation in general:
That is fomula valid in K $\subset$ fomula valid in $S_5$ etc.