If I have a mobile phone repeater installed in my house, can my neighbour use my mobile phone repeater, and if so, will I end up paying for his calls?
No, you will not end up paying for his bills. The billing is associated with the SIM card in your phone. Every phone has a unique SIM card. The mobile phone repeater is a 2-way amplifier. It does not add any information to the mobile phone signal. So, it is the phone that gets billed for making a call – the booster just facilitates you so you can make the phone call.
Solid Walls Block mobile signals.
It is unlikely your neighbor would be able to pick up the amplified signal from your mobile phone repeater. This is especially the case if you live in a detached house. The signal will pass through doors and partitions and it will bounce off walls, but it will not pass through a solid wall. So if you live in an apartment or terraced house, the walls between the apartment or terraced house will block the signals.