Further to sketretal, is it likely that the donor of the aircraft would remove any secure speech system? I have no idea what systems are fitted to Netherlands' Lynx, but in the UK the secure speech is (or was) an overlay on the older clear intercom. Clear was an SG Brown system, Secure a GEC/Marconi. Ripping out the secure speech components would mean the remainder could not be expected to function properly. Here, in the same circumstances, the "secure" part would be removed, but you would be left with components of the clear system that had been modified to anticipate certain interfaces and interlocks.
If the Netherlands use a specific intercom jack, then you may find Tels is connected, but Mic isn't, which would explain some of your problems. In modern (post mid-90s) systems, you may find the Mic/Tels connector doubles as ANR power, in which case yes, the connector must be selected and wired so certain contacts make and break before the others.
Beep is a function of an LRU, not Mic, so it isn't a contradiction to have one and not the other. You may find the Mic Amps and headsets are mismatched.
Sorry, not a lot I can say without schematics. But start at the beginning and use a helmet/mic specified for use in that aircraft.