The answer to the first question is not something that can be provided with absolute certainty. Because there is no discrete "rush" variable present in this expected goal model, if everything about two shots is equal besides one being off the rush and another being off a possession play, then there is no reason that the one off the rush would be higher.
However, it's very unlikely that
everything about a shot off the rush will be equal to everything about a shot off an in-zone possession; the biggest discrepancy will be in the prior event. However, I'm not convinced that the prior events which occur before rush chances are all that much higher than the prior events which occur before in-zone possession chances are actually associated with an increase in expected goal odds. If we look at the factors that drive shot quality, the prior event does play a fairly large role in driving the expected goal value of any given shot, but the prior event has to be one that is
more recent, and the prior event with the strongest influence on increasing xG odds is a shot by the same team; the one with the second strongest influence is a missed shot by the same team. Prior events like giveaways by the opposite team or takeaways by the same team do not even make the list. And a giveaway by the opposing team would likely be occurring with a high time before and high distance from the shot, so I think rush chances really wouldn't change much.
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I think this is actually one of the biggest flaws of the model; that it doesn't properly assess the value of shots off the rush, due to the increased frequency of passing plays occurring before shots off the rush. When the authors of the model wrote about this, they expressed similar concerns:
The full write up is
here, and you can scroll to the very bottom of the page to see the specific plays that they are talking about; 3 of the 5 goals that they consider undervalued are those that come off the rush, with a quick pass occurring before the shot.
So, the answer to your very first question, as best as I can put it, is probably no. It's certainly not a definitively correct statement, which means that the other two statements can't really follow from it.
If anything, you could probably make the argument that if a team is allowing a disproportionately high percentage of their chances off the rush, that expected goals against will
undersell the quality of chances they're actually allowing. Which anecdotally checks out.