The third axiom says that any neighbourhood of a point $x$ is also a neighbourhood of all the points "sufficiently near" to $x$; the reason to introduce it is that axioms 1 and 2 do not put any relation among filters at different points, hence precluding most of the local-global machinery so fruitful in topology.

If I'm not mistaken, you only need $E_G$ to be weakly contractible in order to view it as a universal $G$-bundle, $G$ arbitrary. I guess $E_G$ might indeed be non-contractible for wild $G$.

perhaps do you mean contractible?

perhaps you mean it is not a differentiable manifold.

Now I'm embarrassed, what should I do with my question and your answer?