I have changed it to make it a bit more correct. Technically, it's still only correct if we allow an unbounded message size, but I'm ignoring that in favor of keeping it more readable. If you needed to prove it rigorously for a finite message size, you could instead just use a case distinction (since in the case where there is no winning number, then of course it's not possible to win), but it was never intended to be a rigorous proof anyway, but more as a demonstration of where results like the lottery problem come from.
If I'm following you correctly, then, it would be much easier to just say that each time travel event re-randomizes sensitive events
Kinda. The exact events are fixed for a given transmission though, making it similar to a random oracle over the message space.
Then the odds of transmitting back a winning ticket are the same as guessing a winning number at first.
Yes. But the overall odds of winning are (simplifying slightly) the odds of getting stuck in a loop transmitting the a winning ticket versus getting stuck in a loop that never transmits the winning ticket. (Or getting stuck in a loop where the iteration that receives the winning ticket fails to recognize that fact, etc.)
The better way to win the lottery is to send back every losing ticket, buy one more, and send that one back, too. Eventually, your growing pile of tickets will include the winning number.
In this case, I'm presuming it's not possible to directly send the physical lottery ticket for reasons outlined in 1.4, but if you could, you'd also have to crack into the database that lists all the legitimately-acquired tickets. If you've managed to crack their database, you wouldn't need time travel to win, and otherwise it'd be no better than a forgery (albeit a really, really good forgery).
There is a potential transmission scheme similar to what you describe that is possible; the 'win' condition involves transmitting a circular list of the winning lottery numbers for each world line around in an arbitrarily large cycle. This may actually increase the odds of success. I haven't fully worked out the details yet though, so it's entirely possible that it would perform just the same.
Another way to look at this from a computability perspective, is as a justification of why it's very difficult to construct an Alternating Turning Machine using time travel.