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You're right, it's difficult to sit back and be an armchair psychologist while viewing these people from afar, but you're also right that there are objective patterns of behavior that can be recognized in presidents and raise questions about their mental preparation for the job. Going through my knowledge of presidents:

1. Abraham Lincoln, as you say, is widely believed to have suffered from melancholy or depression. A lot of video documentaries, especially those of the History channel caliber, place undue emphasis on this in an almost conspiratorial way, but Lincoln did suffer from a difficult life. His sympathy for slaves has been suggested to be because of the difficult conditions of his childhood, where his father would rent him out as labor for neighbors and keep his wages. David Herbert Donald also attests to this in his regarded Lincoln. Only one of his sons lived past his teenage years; two of his children had died before Lincoln's assassination, the most recent being Lincoln's favorite son, "Willie" Lincoln, having died in 1862 at eleven. Lincoln's wife, Mary Todd, began to express more of an interest in spirituality (Some even discuss her burgeoning interest in mediums and seances; I am not sure how I feel about the credibility of these accounts.) I have yet to read it, but it's on my soon-to-read list, Lincoln's Melancholy by Joshua Shenk, which discusses his depression in more depth.

2. Woodrow Wilson might not necessarily be relevant as far as "crazy," but I still think he is worth addressing nonetheless. Following the end of the First World War, Wilson faced opposition to the League of Nations (predecessor to modern United Nations) in the senate, primarily led by Teddy Roosevelt ally/mentor Henry Cabot Lodge. Attempting to overcome this resistance, Wilson decided to barnstorm the United States and garner public support; on October 2nd 1919, Wilson suffered a severe stroke that left him incapacitated and in a state that A. Scott Berg more or less hints at as being vegetative. Amity Shlaes, who I generally do not find to be a credible source, also discusses the difficult situation of Thomas R. Marshall, Wilson's Vice President. Although the Constitution allows for a president to be removed in the case of incapacity, Wilson was effectively kept out of sight for the last year and a half or so of his presidency, which was shepherded by Edith Wilson, his second wife, as A. Scott Berg discusses in his phenomenal biography. (Thomas Woodrow Wilson was a widower and remarried as president; he also dropped Thomas from his name as a young man.) One might make the argument we've already had a female president, albeit briefly.

3. Nixon, as you say, also is one of the presidents mostly put into the realm of borderline sanity, and his tendencies did nerve on paranoia, in my opinion. If I remember correctly from his biography, Conrad Black discusses that Nixon admired Mao Zedong and Charles De Gaulle as models for his presidential approach, leading to the so-called Imperial Presidency, where access to Nixon was highly restricted and run through two people, who got to determine who did or did not see Nixon. Black also discusses that Nixon is unusual due to being an introvert, rather than an extrovert president. As Black records it, most every president has been an extrovert, always surrounded by a retinue; in contrast, Nixon preferred to spend time alone in his office with a legal pad jotting notes and introspecting. I believe it is Curt Gentry, in his J. Edgar Hoover, who discusses that, whereas many previous presidents (LBJ and FDR, prominently, though I believe Eisenhower may have as well; not JFK, who Hoover despised, I do not think) might gossip with Hoover or joke about the interesting dirty laundry that came out of Hoover's extensive spying programs. Nixon, unlike these presidents, would listen to Hoover and, dead-serious, ask how the information could be used against the people in question, for Nixon's political gain.

3a) This level of Machiavellian shrewdness and cunning is reflected in Rick Perlstein's Nixonland, which I think makes the most compelling case of all about the mental unfitness of Nixon for the job. Although he does not directly play the role of armchair psychologist, he, like Black, discusses Nixon's distaste for upper-class members of society ("Franklins," from FDR, as you might guess) as opposed to the more run-of-the-mill middle-class Orthogonians with which Nixon identified himself. One might take this a step further and see a parallel in Nixon's later rhetoric referring to the Silent Majority. Nixon's involvement tampering with the 1972 democratic primaries, as Perlstein records it, could be taken as further evidence of his paranoia. Believing he was the only person capable of handling the country's problems, a sort of megalomania, drove him to tamper with the '72 primaries to ensure that McGovern, considered the least viable Democratic contender in the general election, got the nomination; this is a large part of the lead-up to Watergate, as I understand it. Nixon's reaction to the largest strike in US history, in favor of peace in Vietnam, similarly reflects a bizarre thinking: rather than giving credit or listening, he determined to harden and double-down on his position. Honestly, I feel like I could go pull out Nixonland and max out my character count without even getting finished on Nixon.

4. Given the critique of Nixon's megalomania in terms of the "only I can shepherd the country through this" mentality in the above paragraph, it is only fair to point out that by 1944, FDR falls into this category. He was well aware of increasing health concerns, including skyrocketing blood pressure, only exacerbated by his regular smoking and the fact that when he and Churchill were going to conferences like Yalta, they were flying in non-pressurized planes. As Conrad Black, in his FDR biography records, no doctor today would ever endorse a patient, with that kind of health or in that condition, make himself endure that. I believe it was one of his rallies for the '44 election where Roosevelt drove around a rally and was spirited away a few times to take shots to try and keep him warm. He was in pretty bad shape by the end. Again, this borderline does not fall under the category of 'crazy,' but it is an interesting bit of history all the same.

5. Ronald Reagan, according to some, including his son Ron Jr, was beginning to show signs of mental deterioration near the end of his second term. As the article records, this is not accepted by most outside experts or biographers; I am not a fan of Reagan myself, far from it, and I do not place any credibility in claims that by the end of his second term he was suffering from Alzheimer's. While it might be tempting to look at his reaction to things like his militant reaction to People's Park or his comments on MLK Jr.'s death and make an armchair psychologist diagnosis or propose sociopathy, I think that's massively oversimplifying and fails to acknowledge the political tensions and heightened rhetoric of the time. He certainly does show a startling amount of indifference to his family, however, as Perlstein records when he discusses the almost Orwellian way Nancy Reagan ran their household and abused their daughter Patti and the way Reagan reacted to it. Hard to believe he was unaware or did not condone the behavior. As H.W. Brands records, in a well-distributed story, Ronald Reagan was apparently incapable of recognizing Michael Reagan, his son at his graduation; while this story is often repeated, Michael Reagan disputes it. There is certainly an argument to be proposed that Michael is possibly repressing or otherwise rewriting the memory to be more favorable to himself, but that's a whole other can of worms. One might propose narcissism, I suppose, but I think Reagan would likely also be considered tangential at best as a 'crazy' president, I mostly brought him up to dispel the Alzheimer's claim and decided to address a few other possibilities.

Sources:

David Herbert Donald, Lincoln

Daniel Mark Epstein, Lincoln and Whitman

A. Scott Berg, Woodrow Wilson

Conrad Black, A Life in Full: Richard M. Nixon

Conrad Black, Champion of Freedom: Franklin D. Roosevelt

H.W. Brands, Trait to His Class: The Privileged Life and Radical Presidency of Franklin Delano Roosevelt

Amity Shlaes, Calvin Coolidge

Curt Gentry, J. Edgar Hoover

Rick Perlstein, Nixonland

Rick Perlstein, The Invisible Bridge

Will Bunch, Tear Down This Myth

William Kleinknecht, The Man Who Sold the World

H.W. Brands, Reagan