Yes, of course they can throw exceptions. Where did you get that from about not allowing exceptions from no-arg constructors? I couldn't find it in the Java Language Specification.
Also, any superclass constructor can throw an exception. The only limitation is, the subclass constructor can't catch that exception so it must throw it as well.
You don't write a default constructor, so you can't catch it, so it would count as an unhandled exception and (if checked) would cause a compiler error.
Don't get me started about those stupid light bulbs.