For question One, Manafort's claim is that he was charged with working as an Unregistered agent of Turkey prior to the Trump Campaign. The arguement here is that Turkey is not Russia and Manafort wasn't part of the Trump Campaign when the crime was committed. Thus, he would not meet part one of the Order. He might meet part two, if they arose as part of the investigation as outlined in Part one (thus, because he was Part of the Trump Campaign, he still committed a crime under federal jurisdiction, despite not doing it at the time and this was discovered in the course of the investigation in part one of the Order). Part three is basically covering things like obstruction of justice or new items as outlined in Part B of the linked CRF.
At argument is that the Mueller interpreted his order to broadly and that he shouldn't have been looking at matters of concern prior to the campaign. Mueller can argue that if there was a link to Russia it might have occurred prior to the Campaign and was exploited during the campaign, but the counter argument is that Turkey and Russia are not the bests of buds diplomatically.
Mueller is seeking to limit the scope of the investigation, but outlined in the NYT article is that the suit might be a tactic to get to a discovery phase of lawsuits which would force the DOJ to show its hand on what it is looking at with respect to the Special Council. The idea here is a political play that will help further discredit the Investigation as a Political Witch Hunt (as claimed by Trump and Republicans).
This also does not invalidate Manafort's charge in the long run as another non-Mueller prosecutor could attempt to retry him, but this might result in a Fruit of the Poisonous Tree situation, which would invalidate all the evidence collected for this indictment at issue.
From Mueller's perspective it could tip his hand to the public as a good number of possible evidences not yet played would be forced to be revealed and could much more narrowly limit the scope of his investigatory jurisdiction (NYT called it "broad" at the time of Manafort's arrest). As mentioned above, he could lose Manafort as a case due to lack of presentable evidence and any evidence that came from this investigation into Manafort.
The indictment against Manafort would likely have been covered under Order Two, so proving that searching finances related to prior Campaign Work was necessary and how strong that necessity was. I personally doubt that it will fly, but I would like a judge to look over the question.
The liability is really that Mueller will have to show what he's been up to in a more public forum and could restrict what he can look at in Russia. It would also mean that Mueller would lose the indictment against Manafort, possibly evidence collected related to that investigation. This does not mean Manafort escaped a possible criminal charge, just that Mueller doesn't have proprietorial Jurisdiction. Another DOJ lawyer could file charges, but would have to show the evidence was not originating from Mueller (maybe... might fuzzy on this.).