Washington (CNN) Robert Mueller's appointment as special counsel to lead the Russia probe in May caught President Donald Trump by surprise.

He was given no heads up and, according to The New York Times, said, " what the hell is this all about ?"

Since that time, Trump has called the investigation a "witch hunt," suggested Mueller's team members are "biased" and continued to muse about what it might take for him to "fire" the special counsel.

Thus far, Trump has said he "(doesn't) think it's going to happen," but what's to stop him from trying if he changes his mind?

Legal experts say the President's actions raise a host of important questions, testing the limits of what can be done if Trump is determined to bulldoze his way through conventional boundaries.

Here's a look at what is at stake.

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What's the scope of Mueller's authority?

When Deputy Attorney General Rod Rosenstein tapped Mueller to serve as special counsel to lead the Russia probe, he gave him wide latitude.

Not only is Mueller charged with investigating any links between the Russian government and the Trump campaign, but also "any matters that arose or may arise directly from the investigation."

That's why Mueller is allowed to look into Trump's reasons for firing former FBI Director James Comey.

Rosenstein's order also made clear under the regulations establishing Mueller's appointment that he has the specific "authority to investigate and prosecute federal crimes committed in the course of, and with intent to interfere with, the special counsel's investigation, such as perjury, obstruction of justice, destruction of evidence, and intimidation of witnesses."

What this means is that Mueller isn't just looking back in time to what happened on the campaign -- he has his eye on how people behave right now.

Legal experts say it should therefore come as no surprise to learn that his team of lawyers and investigators will methodically gather all relevant evidence -- including seeking a court order to obtain tax returns, if necessary.

Does Mueller have to get Rosenstein's permission first?

Unclear.

"The special counsel shall not be subject to the day-to-day supervision of any official of the department," according to the regulations governing his appointment. However, Rosenstein "may request that the special counsel provide an explanation for any investigative or prosecutorial step, and may after review conclude that the action is so inappropriate or unwarranted under established departmental practices that it should not be pursued."

To that end, some have questioned the extent to which Mueller is fully insulated from interference from the Justice Department, but Rosenstein has tried to put the issue to rest, most recently telling Fox News: "Director Mueller is not reporting to me about individual decisions made in his investigation."

Can Trump fire Mueller?

Under the special counsel regulations, Mueller may be "disciplined or removed from office only by the personal action of the attorney general." Attorney General Jeff Sessions has recused himself from all matters related to the 2016 presidential campaign, so only Rosenstein has the power to fire Mueller.

Rosenstein affirmed as much during a hearing before a House panel in June.

"The chain of command for the special counsel is only directly to the attorney general -- and in this case, the acting attorney general," he said.

Trump does have the ability to fire Rosenstein, for no reason at all, as a member of the executive branch.

As a result, former acting solicitor general Neal Katyal, who helped draft the special counsel regulations, has written that its provisions don't provide a perfect safeguard against others who might want to "stymie" Mueller's efforts.

"Our Constitution gives the president the full prosecution power in Article II; accordingly, any federal prosecutor works ultimately for the president," Katyal explained. "The president, therefore, would have to direct Rosenstein to fire Mueller — or, somewhat more extravagantly, Trump could order the special-counsel regulations repealed and then fire Mueller himself."

What's at stake if Trump tries anyway?

Even if Trump cannot dispense with Mueller easily, the very idea that he might try exposes an inherent flaw in the US Constitution's design, says Harvard Law School Professor Noah R. Feldman.

Feldman points out that the president is ultimately in charge of law enforcement as the head of the executive branch -- a structural arrangement that works just fine until the president or those close to him come under investigation.

So if the President tries to fire Mueller or gets him fired, "it would expose a deep flaw in constitutional design" says Feldman, because it shows the ability of the president to successfully block an investigation -- not a sign of a democratic society.

Trump is "a stress test for this particular constitutional problem," according to Feldman.

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Can Trump discredit Mueller's team?

When Rosenstein appointed Mueller as special counsel to lead the Russia probe, he said he wanted "the American people to have full confidence in the outcome."

But over the last several weeks, Trump and some of his surrogates have sought to discredit certain members of Mueller's team for their political contributions to Democratic candidates

Trump's personal lawyer, John Dowd, dismissed that as " collateral nonsense " in an interview with The Wall Street Journal and Rosenstein told Fox News that Justice Department judges "by results and so my view about that is, we'll see if they do the right thing."

Some legal scholars have also pointed to Justice Department guidelines and federal statutes specifically prohibiting the consideration of political affiliation in department hiring decisions (which apply equally to members of the special counsel team).

Yet Harvard Law School Professor Jack Goldsmith suggests that the "Democratic tilt" of Mueller's team could be a political liability.