In Hidden Features of Java the top answer mentions Double Brace Initialization, with a very enticing syntax:
Set flavors = new HashSet
There's generally nothing particularly inefficient about it. It doesn't generally matter to the JVM that you've made a subclass and added a constructor to it-- that's a normal, everyday thing to do in an object-oriented language. I can think of quite contrived cases where you could cause an inefficiency by doing this (e.g. you have a repeatedly-called method that ends up taking a mixture of different classes because of this subclass, whereas ordinary the class passed in would be totally predictable-- in the latter case, the JIT compiler could make optimisations that are not feasible in the first). But really, I think the cases where it'll matter are very contrived.
I'd see the issue more from the point of view of whether you want to "clutter things up" with lots of anonymous classes. As a rough guide, consider using the idiom no more than you'd use, say, anonymous classes for event handlers.
In (2), you're inside the constructor of an object, so "this" refers to the object you're constructing. That's no different to any other constructor.
As for (3), that really depends on who's maintaining your code, I guess. If you don't know this in advance, then a benchmark that I would suggest using is "do you see this in the source code to the JDK?" (in this case, I don't recall seeing many anonymous initialisers, and certainly not in cases where that's the only content of the anonymous class). In most moderately sized projects, I'd argue you're really going to need your programmers to understand the JDK source at some point or other, so any syntax or idiom used there is "fair game". Beyond that, I'd say, train people on that syntax if you have control of who's maintaining the code, else comment or avoid.