does a broken or blown ac adapter conduct electricity?
Q1: It depends on HOW it failed. It could be open circuit or it could be a short to the input. Or it could catch fire if you plug it in again. It is impossible to tell without debugging it. THROW IT AWAY.
Q2: If you are using these adapters for powering hobby projects, do yourself a favor and go out and buy yourself a proper bench supply. One that you can adjust the voltage and more importantly monitor and limit the current like the one shown below. There are also multi channel varieties.
It might seem expensive, but it will save you large dollars in blown components and circuitry. Or add it to your Santa list.