There is a crack in the wall connecting to the back of the house that seems to be letting black ants into the house, I suspect via the pipework as they come out from ground floor toilet room's sink, maybe the door frame too.

So what can I put into this crack in the wall to fill it up and stop the ants?

Any particular syringe like tool to assist?

Also, what can be causing the mald and paint to be coming off the wall?

Photo 1, your local Home Pro has such accessories, what did you intend filling it with?

I would replaster and repaint.

Is this an extension thats been built on to the house or was it part of the original build?

Photo 2 , its caused by rising damp, google it for causes and treatments.