You can't "be placed" by ATC under any type of service in Class G, except "no type of service".
Yes, we can. It says so in MATS Pt 1 so it MUST be true.
In the case of an approach control service at an airfield in Class G airspace an inbound acft would place himself under control when he first called inbound, otherwise what is the point of calling. This could well be a procedural service and we assume that instructions will be complied with unless the pilot states otherwise.
A departing acft is 'under control' until the pilot no longer wishes to receive a service, or is 10 mins flying time away from the airfield, whichever is the sooner.