Gosh, I should really read more carefully. On page 17 Bowditch answers my QI question. I'll stop now.

And one other thing, as an easy exercise, show that a winning strategy for the angel on the square grid implies a winning strategy for the angel on the triangular grid. (I suspect this is a QI http://en.wikipedia.org/wiki/Quasi-isometry invariant. Perhaps you have to make the angel slightly more powerful...)

Nex: I hadn't played the game yet when I posted the comment :) Didn't realize it was a 1-cat.

This is no longer an open problem. (Well, at least there is a published solution.) It is a win for the angel.

See http://www.warwick.ac.uk/~masgak/preprints.html
and scan down to "The angel game in the plane"

This has also been done for doors with knobs:

http://www.math.brown.edu/~res/Comix/door1.png

Check it out. He does these in xfig.