I'd say you have a problem inside the key switch ... corroded wires/bad connections - the whole assembly is loose or coming apart.
But, that part about pulling the kill switch & the motor still running worries me. That's its sole purpose ... kill the engine when the contact is broken (grounds out the circuit). Now, lots of times it's the kill switch that won't let you "start" the engine ... like when the lanyard clip is not in place (seen/done that). But, never seen a engine keep running when the lanyard clip was removed from the holder. Now, on mine (90 hp Merc) I can simply flip the toggle lever UP and the engine will start/run without the kill switch lanyard clip being in place. But, I've never had the lanyard clip get pulled free and the engine keep running.
If it were me, I'd have the key switch and the kill switch assemblies totally replaced .... unless, of course, I found loose wires, bad crimps, corroded connectors & such, and could do the work myself on the kill switch ... the key switch deal, nope, I'd just replace the whole thing.