If we're strictly going by the nicknames for each presidency, rather than who's necessarily attached to them...
Imperial Presidency - A massive expansion of the authority of the presidency is a likely outcome of this, continuing with the trend from TTL's Civil War, as is a much earlier entry of the United States into foreign adventurism and even colonialism ITTL. Given the possible hints towards an earlier rapprochement with European powers, it's also certain that many will not feel comfortable cozying up to foreign monarchies; republican pride is something deeply ingrained in many Americans.
Busted Flush - I suspect, based on the hagiography that seems to surround him by the present day ITTL, that Kearny is likely to overall be remembered as a good president, maybe even a great one. The usage of 'busted flush,' to me, suggests that someone in his administration that seems promising is likely to squander whatever political capital is left to them or is continually marred by scandal, leaving them with little recourse to continue his legacy.
Best of the Bad Men - Due to the fallout of what might be a one-term presidency, or a continuous humiliation in the headlines for eight years, we're likely to see everyone throw their hat in the ring to run for office and as the title suggests the U.S. will end up with someone who isn't great but things could've ended up much worse in hindsight.
Anger Interrupted - Rather hard to glean what might be meant by this. It could mean anger at some domestic issue - economic downturn is likely, but a war going south is a possible secondary option - that is capitalized on by some firebrand. Given their successor, however, I'm guessing that they never get to do anything and they either die from some illness in office or are assassinated.
Accidental Presidency - Someone ends up in charge who never really thought they'd land in the driver's seat. Pretty self-explanatory.
Velvet and Iron - Given the meaning of the idiom, this is likely a fairly stable presidency and might even catch foreign governments off guard in its willfulness. It isn't necessarily a rejection of the previous administration's positions; it might even be an affirmation of them, whatever they are.
Hangman - Perhaps the most mysterious of them all. While its easiest reading is that of an executioner, it doesn't have to be domestically related. Given the fact we know that at some point the U.S. gets involved in a major war that has them humiliated abroad c. the early 20th century (the conflict in China?), it might be related to the idea of effectively ordering men to their death.