Am I right to assume that in no modal logic, whether in K or in a logic where the accessibility relation is specified as either reflexive, symmetrical or transitive, does ”p implies necessarily p” hold? In what (if any) ways does the accessibility relation between worlds have to be qualified in order for this inference to be provable?
You are right.
If the equation p→☐p is provable, then the only world that is allowed to be accessible from world w is w itself. One could describe that as all worlds being isolated. More formally, this axiom forces the accessibility relation R to satisfy:
If wRv, then v=w.
The whole idea of having other worlds breaks down...
If R is also reflexive (i.e. if ☐p→p is an axiom), you can derive p↔☐p, and the logic behaves exactly as the usual propositional logic with an uninteresting additional symbol.