An effective response to either crisis could allow Trump to demonstrate an ability to govern that has remained elusive during the turbulent first months of his Administration. But for many of the same reasons, scepticism also abounds.
On a high-stakes and closely-watched trip to Texas, Trump seemed to recognise the precarious juncture, as well as the optics, casting the storm in "epic proportions" and emphasising the need to take the long-view regarding the region's recovery.
"We want to do it better than ever before," he said, receiving a Harvey response briefing at a firehouse in Corpus Christi, Texas. "We want to be looked at in five years and 10 years from now, as this is the way to do it."
But in his public remarks he also focused heavily on how his Administration's performance and coordination with local officials is being viewed - "Everybody is talking about it" - while speaking little about the victims of the storm amid worries that Harvey's true carnage will not be known for days.
"There was something missing from what President Trump said - I hope he will say it later today - but that's the empathy for the people who suffer," Ari Fleischer, George W. Bush's first press secretary, said on Fox News.
"In my opinion, that should've been the first thing he should have said was that his heart goes out to those people in Houston who are going through this and that the Government is here to help them to recover from this."
Some Trump confidants have privately worried that his decision to head to Texas before the rain even stopped falling was risky, a premature victory lap that could later haunt him, as Bush learned after praising his FEMA chief for doing "a heck of a job" in the wake of Hurricane Katrina.
The President seemed to anticipate that concern, telling the gathered federal, state and local response officials at the firehouse that "we won't say congratulations".
"We don't want to do that," he said. "We don't want to congratulate. We'll congratulate each other when it's all finished."
On North Korea, Trump warned that "all options are on the table," following the country's missile launch - North Korea's most brazen provocation during Kim Jong Un's five-year-long rule.
Despite the grave warning, Trump's statement - which came more than 12 hours after White House aides had signalled a statement by the President was in the works - was notably measured in contrast to his response to previous tests of ballistic missile launches by Pyongyang. After a recent test, he promised "fire and fury" if the isolated nation continued to provoke the United States and its allies in the region.
This launch, coming after North Korea last month launched two intercontinental ballistic missiles theoretically capable of reaching the US mainland, underscore both Kim's defiance of the international community and his determination to press ahead with his missile programme.
The President's generally rocky start has left him facing a deeply divided, dubious public, with approval ratings at just 35 per cent, according to the Gallup Daily tracking poll.
"In the past, times of crisis has given presidents an opportunity to demonstrate leadership to help solve the issue at hand and help rally Americans behind the president for that and future crises," said Republican strategist Doug Heye.
"The challenge Trump faces is that he has turned off so many Americans with crises of his own making that, even if he handles these flawlessly, he may never win their confidence or even neutralise their unfavourable views of him."
The President's handling of the crises to this point have been, in many ways, trademark Trump. The President followed the storm closely, taking in the television images before taking to Twitter to comment on the "historic," "unprecedented," and "once in 500 year" flooding.
His trip to Texas today - with another one possible on Sunday, to both Texas and Louisiana - was carefully planned, but at times still bore the undercurrents of a political event rather than a disaster relief effort.
In Corpus Christi, Trump was greeted outside by hundreds of fans, who lined the road with signs and chanted, "Texas strong!" and "We love Trump!"
After the storm briefing, Trump emerged and offered an impromptu rally-style speech to a few hundred supporters who had turned out to cheer him on. "What a crowd! What a turnout!" he enthused, before thanking the state's governor and senators. "It's historic, it's epic, but I tell you it happened in Texas and Texas can handle anything."
The President then lifted the Texan flag to loud cheers.
At his second stop of the day, in Austin, the President arrived at the Texas Department of Public Safety to find a sizable anti-Trump demonstration, with signs reading, "Nyet" and "Impeach little hands."
Michael Steele, a former Republican National Committee chairman, said the fact that "we're sitting around waiting to see how he's going to behave or perform" is troubling.
"I think one of the challenges for this Administration is that it has not gotten on course from the beginning," Steele said. "It's going to take a lot more than how you handle a crisis in Texas or how you handle the crisis in North Korea. There's a need for consistency."
With no missile launches during the first three weeks of August, the Trump Administration had suggested that its tough talk toward Pyongyang was working. At a campaign-style rally in Phoenix last week, Trump alluded to his earlier rhetoric on North Korea, telling a boisterous crowd that Kim was "starting to respect" the United States.
"I respect the fact that I believe he is starting to respect us," Trump said at the rally. "I respect that fact very much. Respect that fact."
Those comments, however, came before North Korea's firing of three short-range missiles last week, as well as its latest launch.
Douglas Brinkley, a professor of history at Rice University in Houston, said Trump's behaviour in the face of immense challenges will help determine how he compares to his predecessors.
"A major component of being US president is inspiring the country to pull together in times of cataclysmic events. In the case of natural disasters, the president is supposed to be the personification of care and concern."
Trump's trip to Texas, he said, was a positive, but his "coldhearted behaviour" - including tweeting about his pardon for controversial former Arizona Sheriff Joe Arpaio and plugging Milwaukee County Sheriff David Clarke's book in the midst of the storm - undermined his effectiveness.
"He lost the moral high ground," Brinkley said. "We're not feeling the empathy as much as he's going through the motions."
On North Korea, he added, Trump "complicated his own hand" with his "respect" line during the Phoenix rally. "That's not a sign of respect when you fire a missile over Japan - that's flipping off the president," Brinkley said. "He's so polarised the country that it makes it harder to pull America through a foreign policy crisis."
