The more I think about it, the more interesting these problems get! Problem 1 seems to re-introduce all the issues that CDT has on Newcomb's Problem, but for TDT. I first thought to introduce the ability to 'break' with past selves, but that doesn't actually help with the simulation problem.

It did lead to a cute observation, though. Given that TDT cares about all sufficiently accurate simulations of itself, it's actually winning.

• It one-boxes in Problem 1; thus ensuring that its simulacrum one-boxed in Omega's pre-game simulation, so TDT walked away with $2,000,000 (whereas CDT, unable to derive utility from a simulation of TDT, walked away with $1,001,000.) This is proofed against increasing the value of the second box; TDT still gains at least 1 dollar more (when the second box is $999,999), and simply two-boxes when the second box is as or more valuable.
• In Problem 2, it picks in such a way that Omega must run at least 10 trials and the game itself; this means 11 TDT agents have had a 10% shot at $1,000,000. With an expected value of $1,100,000 it is doing better than the CDT agents walking away with $1,000,000.

It doesn't seem very relevant, but I think if we explored Richard's point that we need to actually formalise this, we'd find that any simulation high-fidelity enough to actually bind a TDT agent to its previous actions would necessarily give the agent the utility from the simulations, and vice versa, any simulation not accurate enough to give utility would be sufficiently different from TDT to allow our agent to two-box when that agent one-boxed.