Sounds like there is indeed some sort of insulation problem along the line, causing both problems you describe. As Jodes said, call in a professional electrician to find the source of the problem before trying to fix anything.

When you have some sort of 'leakage electricity' while connecting the ground, it will trip the rcd (that's the very reason for having the rcds).

The first thing you need to do is find out where the light is fed from.

fed from a lighting circuit:

Putting a 13A socket on a lighting circuit is not expicilty prohibited. However there are a couple of things to bear in mind.

Lighting circuits are normally on a 6A breaker, plugging in a large appliance my trip the breaker.

Sockets intended for general use with a current rating must be RCD protected. Unless the consumer unit was replaced in the last few years the lighting circuit is unlikely to be RCD protected.

If it's possible to get a cable back to a socket circuit without unreasonable effort then that is what I would do. If not I would fit the socket on the light circuit but clearly label it with something like "burgler alarm only, fed from lighting circuit".

If you do go for the socket circuit option remember to follow the normal rules for that (which depend on exactly what type of socket circuit it is)

fed from a socket circuit via a fused connection unit:

This is similar to above except you will be more likely to have RCD protection in place.

fed from a socket circuit without a fused connection unit:

You need to add a FCU to procide overcurrent protection. Cable from the socket circuit to the FCU should be at least 2.5mm². I would advise fitting a 5A fuse though you may be able to fit a 13A one depending on the type of light fitting and the size of any wiring.