Not sure on the buzzing, never heard of that one. I would think there might be a switch in the ignition lock which may be causing this. I used to work on this chassis a lot but never heard of this, but it's been awhile. Are you sure the lights are turned off? It's not giving you any warning in the cluster as to why it's buzzing?? Like "Take Key" or "Headlights On"???
On the locking, if you are sure the synchronization was done properly, then I'd assume a bad key, my first thought also was the batteries, make sure they are installed properly and all the plastic film was removed from them, sometimes they protect them with adhesive tape.
The "wrench" is simple, that's the FSS telling you it's time to do a service, it says in the owners manual how to reset this you can do it yourself, look under FSS or Flexible Service System in the manual index.
Gilly

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