Problems 1 and 2 both look - to me - like fancy versions of the Discrimination problem. edit: I am much less sure of this. That is, Omega changes the world based on whether the agent implements TDT. This bit I am still sure of, but it might be the case that TDT can overcome this anyway.

Discrimination problem: Money Omega puts in room if you're TDT = $1,000. Money Omega puts in room if you're not = $1,001,000.

Problem 1: Money Omega puts in room if you're TDT = $1,000 or $1,001,000. Edit: made a mistake. The error in this problem may be subtler than I first claimed. Money Omega puts in room if you're not = $1,001,000.

Problem 2: $1,000,000 either way. This problem is different but also uninteresting. Due to Omega caring about TDT again, it is just the smallest interesting number paradox for TDT agents only. Other decision theories get a free ride because you're just asking them to reason about an algorithm (easy to show it produces a uniform distribution) and then a maths question (which box has the smallest number on it?).

You claim the rewards are

independent of the method that the agent uses to choose

but they're not. They depend on whether the agent uses TDT to choose or not.