There is an empty lot next to my house. A block wall separates the properties. Someone has made holes in the wall (on the empty lot side) for no apparent reason and it appears to have weakened the wall. There are cracks in it now and it is leaning towards my house. It is falling. It is sure to fall onto my house eventually. How do I find out who the owner of the property is?

Get in touch with your local council.
That is where you pay your rates.

I live next door to an empty lot. There is a sidewalk that goes through to the next street, but the street does not go through. There are half dead large trees in the lot and one of them is hanging over my yard and making a mess. Can I legally have this tree pruned so that it no longer hangs into my yard?