I don’t understand the special role of box 1 in Problem 2. It seems to me that if Omega just makes different choices for the box in which to put the money, all decision theories will say “pick one at random” and will be equal.
In fact, the only reason I can see why Omega picks box 1 seems to be that the “pick at random” process of your TDT is exactly “pick the first one”. Just replace it with something dependant on its internal clock (or any parameter not known at the time when Omega asks its question) and the problem disappears.
This is a strategy that I sometimes implement and I think it should be used more often, be it only because it allows friends of friends to meet or get to know one another better and that makes nicer social networks.
However, I wonder if it can be taken too far. If everybody routinely makes this sort of demands, won’t people stop to answer them ? Is this technique successful partly because it is not widely used ?