1. Yes, there is a connection, as you point out. In the Tractatus, Wittgenstein writes:

3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

Gödel, as you know, proceeded to do precisely that.

2. Wittgenstein's argument against type theory is one of many factors that changed Logical Positivism. Russell's "barber paradox" was another. If the history of Logical Positivism interests you, I'd recommend a delightful graphic novel called Logicomix which covers the territory nicely.

3. Not really.

1. Yes, there is a connection, as you point out. In the Tractatus, Wittgenstein writes:

3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

Gödel, as you know, proceeded to do precisely that.

2. Wittgenstein's argument against type theory is one of many factors that changed Logical Positivism. Russell's "barber paradox" was another. If the history of Logical Positivism interests you, I'd recommend a delightful graphic novel called Logicomix which covers the territory nicely.

3. Not really.